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Win probability added

From Wikipedia, the free encyclopedia

Win probability added (WPA) is a sport statistic which attempts to measure a player's contribution to a win by figuring the factor by which each specific play made by that player has altered the outcome of a game.[1] It is used for baseball and American football.[2]

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  • ✪ How Understanding Probability Helps Us Make Better Decisions with Mehran Sahami
  • ✪ Expected Value
  • ✪ Odds Ratio & Relative Risk Calculation & Definition, Probability & Odds

Transcription

[MUSIC] >> [APPLAUSE] >> Thanks very much. And welcome back to the farm. Like was mentioned, this is a hard place to leave. [LAUGH] So much so that some of us never get escape velocity to actually go very far. And I'm looking forward actually, next year is my 25. So I'm looking forward to coming back, although I'll already be here. So what I wanted to do was spend a little bit of time taking about probability and decision making. And sort of as we go along, I may ask for some interactivity on your part as well. So, as you know, decision making is not a spectator sport. And so along the way they'll probably be some decisions we need to make. One of the things, I have a very high level of respect for alumni. So I will try not to pelt you with candy unless you want to be. The students I have actually like it when I throw it out to them. >> [LAUGH] >> But I figure in this, you're a well known crowd, I probably shouldn't [LAUGH] necessarily be throwing things, but just so you know. What I wanna spend a little bit of time doing just to start is to kinda frame the problem. And then we need to talk a little bit about some notation, get into a little bit of math. And then talk about what that really means when we start thinking about decision making. So a lot of the decisions we make involve uncertainty. In some sense they have to involve uncertainty because if they don't involve uncertainty and we don't have multiple choices to make, we really don't have a decision at all. So, the times when we think about having a decision to make, it's because we have options from which to choose. And we have some uncertainties surrounding what might be the outcomes with respect to those options. And so, part of the idea is, could we actually use that fact? When we think about that uncertainty could we quantify it in a way that will allow us to make better decisions? And so probability's gonna be the mathematical tool that's gonna underlay what we do. The basic idea is that it's just a way to formally quantify uncertainty. If we don't know when something's gonna happen, we wanna be able to put some chance or some probability on that. And that gives us a way to be able to reason more formally about it. And so when we think about the future and there's uncertainty in the future, this is gonna be the notation we use to think about that uncertainty. And if we can quantify it then maybe we can make better decisions. And so some of you may know the decision analysis is a field that existed for a long time. The term decision analysis was actually coined by our own Ron Howard here over 50 years ago. And basically, it's a normative process for decision making. So, what I mean by that, its the way decisions for some definition of should, should be made. It's not necessarily how they actually are made. And we'll talk about that. People make irrational decision for some definition of irrationality all the time. But this notion of the kind of decision we should be making and the principle that underlies it is a notion of maximizing expected utility. To understand that notion, we need to understand what utility is, what expectation is, and then how to maximize it, and so we'll talk about all those things. But one thing to also keep in mind is not all decisions are appropriate for formal analysis. That's something that Ron always liked to say too. The decision of who to marry is not necessarily one that you do formal analysis on. I don't know. Is he gonna lose his hair 20 years from now, let me write down a probability for that. Not something we think about right? So there are some decisions for which this is not suited for. But there are many sort of, we could think of as rational decision making places were this framework could work. So, let's talk a little bit about some of the underlying math to kinda get started. And by the way, I should mention, if there's any questions or anything's unclear along the way just feel free to ask a question, yell it out or raise your hand. It's much easier to make sure that everything's clearer as we go along rather than waiting until the end for questions. So, first thing, if we wanna be mathematical, one thing mathematicians always like to do, is they like to give highfalutin names to really simple concepts. So it looks like what we do is more complicated than it really is. Okay, so the first thing we're gonna do, and by the way I brought my trusty lightsaber. Cuz I grew up in the Star Wars generation. And I just think it makes for a nicer pointer. So that's really all it is [SOUND] but it does turn on. >> [LAUGH] >> So if anyone gets outta line, [LAUGH] I got the light saber. So a sample space is the first concept we talk about. And a sample space is just a term for a very simple idea. It's the set of all possible outcomes of an experiment. What does that mean, what is an experiment, what are outcomes? So if I take this thing, it's my big six-sided die and I roll it, that's an experiment. I don't need a white lab coat. I don't need an Erlenmeyer flask. All an experiment is is something for which until I do it, I don't know what the result is. So when I roll this die, until I roll the die, I don't know what the outcome is. But there's a set of possible outcomes that could happen. The set of all possible outcomes that could happen is what's known as the sample space. So if I flip a coin, I also brought my giant coins with me. If I flip a coin, there's two possible outcomes, heads and tails. So that becomes the sample space is the set of all possible outcomes. If you're one of the old Twilight Zone fans and you're like, what if it lands on it's side and then I can read minds. That never happens, okay? So we just have heads and tails. If I flip two coins, I can think of the silver coin and the gold coin. Not real silver and not real gold. But, these are separate coins. So if I flip both of them, I can think about four possible outcomes. Namely, I get heads and heads. I get heads and tails. I get tails and tails. Or, I get heads and tails in the other direction. So it matters which coin for example the outcome happens on. And we just refer to those as pairs, so we have these four pairs up there. Rolling a six-sided die has six possible outcomes. If I roll two dice and I ask for the sum of the two dice, then the possible values that can come up are the numbers 2 through 12. So the other concept we need to think about is something known as an event space. An event space, again, is a simple concept with sort of a highfalutin definition. The formal definition is it's the sum subset of the sample space. What does that actually mean? What that means is, of the outcomes that are possible, what are the ones we care about? So if I flip a coin and I say, care about the coin coming up heads. Then the event space is heads, which is a subset of all possible outcomes, heads and tails. If I flip two coins, and I say I care about getting at least one head. Then I consider all the possibilities where there is at least one head. Namely, I could get two heads, I could get a head and a tail with the heads on the gold die. Or on the gold coin. Or I could get a tail on a head where the tail is on the gold coin and the head is on the silver coin. But tails tails is not a possibility that satisfies the event I care about. If I care about rolling an odd number, it's the numbers one, three, and five. If I care about rolling a number greater than 8, it's just 8 through 12 on the two dice. So that's our event space. And so when we think about that, there's a set of things that could happen, there is some set that we care about. Then it becomes a question of determining how likely are the things we care about. And so some things have what are known equally likely outcomes. So if I role a die, if I assume this die is fair, then all possible sides one through six are equally likely. And that happens a lot in the real world. So with flipping a single coin, with flipping two coins, assuming the coins are fair, rolling a six-sided die, all the outcomes are equally likely. And so the probability, which is the notation I'm gonna use here, p of a given outcome, is just 1 over the number of outcomes. So if I roll this die, the chance of rolling a five is one sixth, cuz there's six possibilities. The chance of rolling an odd number is three over six cuz there's three possible odd numbers out of six possibilities. And so that generalizes. In the case where there's some event e that we care about, like rolling an odd value, we just look at the number of outcomes that satisfy the event we care about divided by the total number of outcomes through the size of the samples basis we say about it. And so that defines in a special case where it with equally likely outcomes the chance of different things happening. So as I mentioned, flipping heads has a 1/2 chance of happening, rolling a 5 or 6 is 2 out of 6 possibilities, or 1/3. Now, where this begins to get interesting is we can begin to look at some examples where things become counter-intuitive. So I'm gonna give you some classic statistical problems that sometimes people bring up. But if you've seen these before don't yell out the answer, if you haven't, we'll go through them. So one of the classic ones is birthdays. What's the probability that if I have n people in a room, so you are n people for some number n, which is probably I'd say around 100, maybe 150. What's the probability that none of you share the same birthday? Now that's regardless of year, so that's our first caveat, so it's actual, just calendar day. And we're going to rule out leap years. So just wondering, anyone born on February 29th? No one, shocking. Because I have a son who was almost born on February 29th, and the doctor told us, if he's really born on February 29th, you can either pick February 28th or March 1st as his birthday. So it turns out, there's actually very few people, for many reasons, there are very few people who are born on February 29th. So if we use this little bit of mathematics that we've seen so far, how can we figure out the chance of this happening? So if the number of ways to have birthdays for n people, each of the people can choose any one of 365 days to be their birthday. So if we want to say, what are all the different ways we could set birthdays in this room, each of the n people can pick one of 365 days. And so the number of possible ways to specify birthdays is 365 to the nth power. So how many ways can we specify a birthday such that no two people share the same date? Now we create a constraint, because the first person, if we were to start here, you get to pick any one of 365 days. You can pick any one of the days except for the one he picked, so that gives you 364 options. You have 363, 362, etc. And so as you can imagine if we had 366 people, there's no way we wouldn't have one that wouldn't share a birthday. But basically what we get is this little series that each person has one fewer option, and we multiply all the options together. So our probability is basically just what we refer to as the event space, the way we can set all the birthdays that satisfies the condition we care about. Namely, that no one shares a birthday, that's the numerator. The denominator becomes all the ways we could specify birthdays, that's our sample space. And so we get this interesting little fraction that the bottom and top both have n terms that we multiply. The one on the bottom is 365 and it remains that. The one on the top just decreases by one every time we do the multiplication. When we see something like that we might say, how many people that we need to have before we share a birthday? So I'm gonna do a little experiment in here. I'm gonna start with a month, everyone whose birthday is in that month, raise your hand. And then I will call on you and you yell out which day of the month is your birthday, and if there's someone else in the room that matches, yell out match. And I'm willing to bet, in this room we have a lot of people but we have maybe half the calendar covered, that we are virtually guaranteed to have two people share a birthday. So January, all the Januaries, hands up, what day? >> 7. >> Anyone January 7? No, all right, next January? >> 26th. >> Next January. >> 30. >> Third? No one's a match yet? >> 12. >> 12, 18 any other Januaries? >> 15. >> 15. >> 1st. >> 1st. >> 8. >> 8. Any other Januaries? We could actually make it a month without any, way back there. >> [INAUDIBLE] >> Which Day? 9th, all right so no Januaries, amazing. Now let's do February, all the February hands up. Date? [INAUDIBLE] >> Match. >> Match, so we finally got a match. So we got to February, which I'm actually astounded we even got to February. And the reason why is if we looking at these numbers, the chance of there being two people with a matching birthday, if we have 23 people in the room. That's a tiny fraction in the whole calendar, that's less than one-tenth you could think of as the size of the number of days in a calendar. And at that point, there's a 50% chance already two people match, okay. If we go up to 50 people, it's now less than 3%. If we go to 61 people, we get to less than half a percent. And we get to 80 people, we're already down to one one-hundredth of a percent, all right? So in this room, it was near guaranteed. With 150 people, it turns out it's roughly about one over three times ten to the 17th power, okay? So I was almost willing to put money on it, but there's still a non-zero chance. >> [LAUGH] >> And I was like okay, there's a small chance it couldn't, I actually got worried when we got through January. So it's somewhat counterintuitive that this would be the case, but it turns out if you look at the problem more closely, it's not that counterintuitive at all. And the reason is here's the first person, they pick a birthday. Here's the next person, they pick a birthday, and we compare them. Then when we have the next person, we compare them to both of the others. And with the next person we compare them with three others, and the next person we compare them to four others. And you can see quadratic growth in the number of comparisons we make as we add more people. And so that's why this number drops off so quickly in terms of the number of people relative to the actual size of the calendar. But if we take this same concept and tweak it just slightly, the problem looks very similar, but the dynamics completely change. So now let's consider the probability that we have n people in a room, so we'll say n people other than me, what's the probability that someone in here shares the same birthday as me? Okay. So if we wanna think about that numerically, we'd again say how many ways can we assign birthdays to everyone in the room, 365 to the nth power, because anyone can have any birthday. How do we assigned birthdays to people in the room such that no one has the same birthday as me is 364 to the nth power, because everyone can pick a day except for my birthday. And so the probability that we would satisfy the condition that no one has the same birthday as me is just 364 to the n over 365 to the n. It look super similar to what we had before, right? The denominators are the same, the only difference is that numerator sort of remains the same every time we multiply it, rather than decreasing by one. That completely changes the problem dynamic. So, it turns out now if we have 23 people, which before gave us a 50% chance of matching birthdays, in this setup there's a 93.9% we have no matching birthdays. At 100 people at 76%, we still have no matching birthdays. 250 people is what we need to get to before there's a 50% chance of matching birthdays, which seems counterintuitive. Because you ask most people if you had a calendar, how many people would you need to have a 50% chance of someone matching a birthday as you? They'd probably say, around 183, about half the calendar. It turns out that's not the case, because people can share each other's birthdays. So here is you, first person comes along, we compare them to you. Next person comes along we compare them to you, next person comes along we compare them to you. The only issue is, these people can share the same birthday and there is no issue with that. So we don't have to get, we need more than 180 people to get half because some of those people actually share birthdays. So just ask. So that's why these probabilities are so much higher than before. Anyone's birthday May 10th? We actually got one, come on down. >> [LAUGH] >> That's my birthday. >> [APPLAUSE] >> I've never had that happen before. >> [LAUGH] >> So, if, actually you know what? I'm gonna give you a candy right now, you're gonna be a volunteer for our next experiment that we're gonna do. Okay, so if you want to just hold on right in the front row. That's amazing. So, the chance of it was, with this size crowd, there was actually probably around 50% chance it would happen. But I've never actually had that before, but they were smaller crowds. Okay? So the next thing we want to think about, so now we get a little bit of the notion of just trying to understand the problem dynamics we need to pay a lot of attention to. The next think we think about is what information do we have available at the time we make a decision, okay? And so the idea is there's this notion of conditional probability. And the idea of conditional probability is what's the chance something happens given that we have already observed something else has happened that might in some sense change the chance of the thing we care about? That's referred to as conditioning on f. If we wanna speak probabilistically. And the way it's written is, we say, what's the probability of E, and we have a little bar and F as after it. F is just some other thing we've observed in the world and the way we would say this is, what's the probability that E will happen, some event we care about, given that we've already observed something else called F, okay. So there's this little theorem, the details in the theorem are not important, we're actually gonna use this theorem to do a couple of things in just a bit. But there's a theorem that's due to Thomas Bayes that we'll talk about that basically says, if you have this probability, what's the chance of E happening given that F has already occurred? Through some basic manipulations, you could actually flip around that probability say what's the chance F will happen given that E has already occurred? And you might say, but that's a little bit weird cuz I've observed F, and I care about E. Well, sometimes it's actually easier to think about what if E were true, what would the chance of F be? Doctors do this in diagnosis all the time. I wanna see if you actually have some particular medical condition. I take a test, and so I say what's the chance you have the medical condition if the test were positive? But it's easier to flip it around and say what's the chance the test is positive if you actually had the medical condition. So that's the thing that's easier to get from data from people we know have the condition or not got the condition. But we need this rule to be able to turn it around to actually diagnose people. And it turns out this becomes super useful for decision making, okay. So just to give you a simple example, I could say what's the chance or rolling a 10 on two dice? Well, if these are distinct dice, the chance or rolling a 10 on two dice turns out to be 3 out of 36, cuz there's 3 combinations of the dice. Namely a 4 and a 6, and a 6 and a 4, and 2, 5 which are 3 out of 36 possibilities that would give you 10. But if I know one of the dice already rolled to 5, then all I care about is this die coming up with a 5, which has a one-sixth chance, so it becomes different when I've observed some information about one of the things involved. And so I could get that from Bayes' Rule by saying, what's the chance of rolling a 10 and a 5 on the first die, that would be getting a 5 here and rolling a 5 here. There's only a 1 in 36 chance of rolling that on two dice if I'm rolling them from the beginning. And then divided by the probability of getting the thing I believe I have previously observed, which is getting the 5, okay. So let's try an example with this first, I'll tell you about Thomas Bayes, in case you're wondering. Thomas Bayes was a reverend that lived in the 18th century, British mathematician and Presbyterian minister. He also looked remarkably like Charlie Sheen. >> [LAUGH] >> That's kind of the young svelte Charlie Sheen, we won't talk about the later Charlie Sheen, okay? So where this gets interesting is we can begin to apply this to some problems that people actually do, and then we'll do a little experiment, and one of them is text analysis. So here's an interesting concept, what if we took the words that someone had written and ignored the order in which they were written. And you would say, but that doesn't make any sense, why would I ignore the order of words, trust me if you've read some student essays, they ignore the order of the words too. >> [LAUGH] >> But let's just pretend. And if we ignore the order of the words, what's the probability of any given word you write in English? Well, how do we do that? We could take a piece of writing that someone has written. Basically, we could cut it up on little slips of paper so every word's on a different slip of paper. Put them in a big barrel, shake it up, put our hand in, and draw out a word. What's the chance we draw to particular word like the or chair? It depends on how many words are actually in the barrel, and how many times that particular word is used in language. So a simple example would be the word the is much more commonly used than writing than the word transatlantic. So we'd expect the probability of the word the to be higher. Here we'd like to think the probability of word Stanford is higher than the probability of the word Harvard. There's [LAUGH] nothing wrong with that, okay? So we can measure the probability of each word for someone's writing, but here's where things get interesting. Now, we can think about the different word usage for different kinds of authors. So someone, for example, given that someone lives in California, the chance they would write the word sunny is probably higher than someone who might live on the East Coast. That's not to denigrate anyone on the East Coast, thanks for visiting California. It's just in writing, that's what tends to happen. So if that what tends to happen, if you say given the writer, that tells me the probability of particular words. You then think back to our friend Thomas Bayes who said if you have one probability, I'll show you how to flip it around. Which means now if I don't know the writer, I only know the words, I can determine the probability of someone being the writer. Which has actually been used to deanonymize many things that have been written anonymously in the past. Let me give you an example. The Federalist Papers. Remember the Federalist Papers? I don't know, like American history somewhere back in high school, it was a fun time. Just to refresh your memory, the Federalist Papers were 85 essays that were written about the ratification of the Constitution. They were all written under the pseudonym Publius, turns out there was three people, Alexander Hamilton, James Madison, and John Jay who comprised Publius. But they had written a bunch of other stuff that luckily for us, they'd actually put their name on. So we could go to these fine folks' actual known writings analyze the distribution of words that they used and then use Bayes' theorem to figure out who wrote which essays. And we could actually also figure out that there were a couple of essays that two of them collaborated on because the distribution of words used was actually a combination of two different people. I like to tell my students this because sometimes they like to post stuff anonymously online, and then they get really disturbed when they see this, okay? >> [LAUGH] >> It turns out doing this kind of analysis of word usage by people is also something that you may encounter every day in technology that you use, for example, spam filters. It turns out, in a spam filter, a lot of people who sends spam, are advertising things like Viagra. The chance for the word Viagra appears in an email that was written by you, generally I would imagine, is smaller than the probability that it appears in a spammer. And if you ever wonder how your email system figures out what's spam and what's not spam, it's actually using a system like this to determine probabilities of words and messages that are sent to you and then figure out should they go on the spam folder or not. So there are decisions being made using this kind of technology whether or not they're fully explicit or not. And just to show you this little graph, this is from Gmail if there's any Gmail users out there. The orange line is the percentage of all emails sent through Gmail that's spam. So this is a slightly historical graph cuz I can't show you the latest numbers, but that's approaching 80%. It actually turns out the amount of all email that's sent that's spam, generally, is estimated to be about 90% now. So just wondering anyone get nine spams for every one real email message you get? If you do, you should change your email address, okay. But if you don't, the reason is that blue line is the percentage of spam that actually makes it through the system, that's not detected. Here is the interesting thing, as the amount of spam goes up, the percentage of spam that gets through goes down. And the reason for that is that we have more data for spam, we actually learn the characteristics in the word usage in spam better, and we can actually do a better job filtering it out. Okay, and this is actually used with a bunch of different factors, okay. So now we're gonna play a game for which I have my illustrious volunteer. We're born on the same day, which is very exciting for me. So there was this game show called Let's Make A Deal. Anyone remember Let's Make A Deal? So back in the day, yeah. When I say let's make a deal to my students, they're like what are you talking about? >> [LAUGH] >> But luckily, some folks remember Monty Hall. So here's how the game works. There's three doors in the game. Now it turns out they wouldn't let me install three doors in this room, so I have three envelopes. And so what the host would do. We have these three doors, we'll call them A, B, and C. Behind one of the doors or inside one of these envelope is a prize, and the other two are empty, okay. And so what the host lets you do is basically pick one of the doors, in this case, it will be one of the envelopes. So you can choose one of the envelopes. Feel free to pick whichever one you like, okay. Don't open it yet. Now, all of them have a Green sheet of paper and there so is you don't go like this and try to figure out what's inside. But one of them also contains a portrait of Andrew Jackson known as $20 bill, okay? And so now, here's the interesting thing what Monty Howard doing that makes could deal, is Monti would open one of the other two doors. Right, so Monty opens one of the other two doors, here's the green slip of paper, and there's no money in there. And then what Monty would do is say all right, there's one other door that's available. Would you like to switch for that door? Okay and so the question is, does it make sense to switch for that door? Okay so should we, how many people say yes because you took a statistics class, how many people say no, a few people? How many people are in that indeterminate it's the afternoon on a Friday I shouldn't be making decisions, good? >> [LAUGH] >> So let's look at this problem now with all these tools that we've gotten to figure out what's actually the right answer, is there a right answer. If we don't switch right so if you keep the envelope that you have, there is a one-third chance you win. Because a prior you were given no information you pick one of the envelopes, let's say at random. And assuming you don't have psychic abilities, you have a one-third chance of winning. Now let's look at the possibilities. So now to be sort of mathematical we need to give everything some letter designation. So we're gonna call the envelope you picked A, so that will be A, okay? So now, there's three possibilities, one possibility is that A is actually the winner. The chance that A is the winner is one-third, that's the chance you picked the winning envelope. And what would happen in that case? If you picked A, and A was the winner, I would be holding B and C which are both empty. So I could open up either one and reveal to you an empty envelope. And then if you were to switch, you would be switching for another empty envelope and so you would lose. And so the probability you win if A is the winner and you picked A but then you switched to zero cuz you're switching to an empty envelope. Now it doesn't sound so good. So let's consider the other possibilities. Let's say you still have A, so the envelope you have is always A. Let's say B is actually the winner now. So B has a one-third chance of being the winner. So what does that mean? You have A, I'm holding B and C and B is the winner. So I'm forced to open C, cuz if I open B and say here's the money, you wanna switch? People don't like that game, so what we're gonna do is I'm forced to switch to open C because that's the only envelope I have that I can show, you're holding A and I'm holding B, which is the winner in this case. So if the probability you win, given that B is the winner, and you chose A but then you switch to a guarantee to switch to B and win. So the probability's one of winning. And by symmetry if we consider the case where C is the winner which also has a one-third chance you're holding A, I have B and C. If C is the winner I'm forced to open B which means if you switch, you switch to C. And so now what we need to do is add up those three possibilities cuz those are the three possible worlds that exist given their probability. So there's a one-third chance A is the winner, and then if you switch you get zero or the probability of winning is zero. There's a one-third chance B is the winner, and if you switch your probability of winning is one. And there's a one-third chance C is the winner and if you switch, your probability of winning is one. When we add those together we get two-thirds, so would you like to switch? All right, all right, go ahead, now, you can open it. I hope this works. >> [LAUGH] >> Yay. >> There you go, Andrew Jackson, you can hold it up for everyone to see. >> [APPLAUSE] >> So thanks for all those years of tuition, you do get something back. >> [LAUGH] >> So sometimes even then is hard for people to see the solution for this problem, which is actually not out of the ordinary. As matter of fact, this problem originally, it has much more, longer statistical origins, but Marilyn vos Savant was a woman who actually had an advice column called Ask Marilyn. And she posed this problem in there and she said, you should switch. Because it was a time when Let's Make a Deal was on TV. And a whole bunch of people, including mathematics professors, wrote to her and said, no, you're wrong. How can you possibly do this? And it turns out that Marilyn vos Savant was actually in the Guinness Book of World Records for having the highest IQ of any woman. She was right. So this story itself actually has some historical context to think about the fact that when it's between math and intuition and they collide the math wins. So here's a slight variant to clarify things. What if instead of three envelopes, I had a thousand envelopes and I let you pick one, okay? The chance you choose the winner is one in a thousand. But what would happen is, there's nine hundred and ninety-nine envelopes over here and one over here. This one you pick has a one in thousand chance, which means, of the remaining envelopes, there's nine hundred and ninety-nine out of a thousand chance that the winner is over there. Okay, what the host is forced to do is open 998 of the envelopes, so all the envelopes except for one, which means if the winner came down this side there are 998 envelopes that are empty that have to be opened and then the winner remains. What's the chance that the winner came down this side, 999 out of 1000. So in that case after opening those 998, the probability of switching and winning has to be, this probability has to remain because this was the chance the winner came down the side. It just makes it more difficult with three envelopes or three doors because we have a situation where we have a one-third probability you picked a winner and two-thirds that the winner came down this side. So with smaller numbers it's harder to see, but it's exactly the same problem. Question? >> It says it's conditional on the person running the test, knowing which envelope it is. >> Yes, yes, yes, that's a great, we should chat more, are you a statistician by the way? >> No. >> No, all right but there is an underlying, what is the actual mechanics of the problem at play. That's absolutely true and for that I will lay this at your feet, because you know the subtlety and the problem. >> Thank you. >> Could I ask that same question on the one you first did, which I'm still trying to work through cuz I don't know the TV show. >> Okay. >> The reason it comes out the way it does is because the host has to know which is which, and so it's not really random. >> Exactly, the host- >> He's sort of revealing something when they [INAUDIBLE] >> Absolutely, so the issue is that the host is constrained to open an empty door. The host is not picking randomly. >> [INAUDIBLE] to do that on the show, I'm just curious now. >> Well I'll tell you when I was a kid and I used to watch that show, I used to always sit there going don't switch, don't switch, don't switch. And I didn't actually count but I did this demonstration for some high school students and much to my shrug grin, the student who picked the envelope. All the other students in the class said don't switch, don't switch and I got these shades of when I was a kid and then it turned out he didn't have the winner anyway. So you know the math and intuition collide, the Math wins and I told him that but he was pretty sad. >> [LAUGH] >> So should you switch the chance of winning is one over the number of original envelopes. And if you switch it's the number of original envelopes minus 1 divided by the total number of envelopes, okay? And it just turns out the one-third, two-third case is sometimes harder to see that. So one of the other things we can think about in terms of decision making is when we wanna make a decision, what do we expect to get from that decision, okay? And part of the ideas what we refer to as expected value which is a statistical concept. Let's say we have some variable that we're gonna call x, which represents the result of rolling a die, six-sided die. And we wanna say what's the expected value. The expected value is basically just a way of saying what's the average we would expect. So we have some probability of different outcomes. We take every possible outcome, that's little x. So there's, all the little x's are all the little possible outcomes one through six and we multiply each one of them by the corresponding The corresponding values, 1 through 6, by the chance of that value coming up, okay? So I'll show you this explicitly for a six-sided die in just a second, but that's the notion of expectation. You could also think of it as average, or the weighted average, if this was not a fair die, if the different sides came up with different probabilities, okay? So it's also referred to as the mean, which some of you may know from exams back in the day. And the way we could compute it is, we basically say for a six-sided die, the chance of any outcome, 1 through 6, is all equally likely, is one-sixth. So what's the average value of a roll? The average value of a roll says, take the chance of rolling a 1, multiplied by 1. Add to it the chance of rolling a 2, multiplied by 2, the chance of rolling a 3, multiplied by 3, etc. In this case, all the probabilities, the chances are the same, but what we get is seven-halfs or 3 and a half. Which is weird because we can't actually roll a 3 and a half. But if you think about us rolling this over and over, what's the average value over all those roles, is 3 and a half, okay? So, that's expectation. Now we can take this and apply it to situations were there's some uncertainty and some payoff, it's not just a die role. We could say, let's say I have a coin to flip, and I say, if this coin comes up heads you win $5, and if it comes up tails, you lose $2. [CROSSTALK] And it's a fair coin. >> There you go, someone knows the probabilities involved. Sorry, I'm also really bad shot. So we can define this, look at this mathematically, and say let Z be our winnings, okay? What's the average winnings of playing this game? And so if we think about the average winnings, what we say is it's one-half the probability of getting heads, times the outcome, which is we win $5. Plus one-half the prob, which is the probability of getting tails, times the outcome when we get tails, which is negative $2. And we just multiply and add, and we get the average payoff of the game is positive $1.50, so it's worth playing. >> So if you wanna lower the variance, you can just pay me a buck 50 right now, and we'll call it even. >> [LAUGH] Yeah we can play over and over, if I brought that kinda cash with me. But we will play another game in just a bit. And so we can begin to think of this idea, to think about different decisions and the possible outcomes, the possible payoffs, by using what's known as the probability tree. And a probability tree is just a graphical way of representing a decision. So we could say, flipping a coin, there's two branches, there's two possibilities. The coin with probability p comes up heads. And with probability 1- p, it comes up tails, cuz the two probabilities added together have to add up to 1, okay? So, let's consider the decision to play a lottery. So what we can do is, we can say do you buy the ticket or not? You have a choice, yes or no. If you don't buy the ticket, you don't win anything, you don't lose anything. If you do buy the ticket, then you're entered into the lottery. There is some probability p you win the lottery, some probability 1- p you lose the lottery. And if the lottery pays off $1 million, in the case where you win, which has probability p, your payoff is $1 million- 1. Why is there a- 1? >> [INAUDIBLE] >> We gotta buy the ticket, right? Tickets are not free, so we- 1 for the ticket, and if you lose the lottery, so you decide to play but the outcome is that you lose, you just lose your dollar. And so the question is should you play or not, you can look at what is the expected value in those two outcomes. Your expected value, there's no uncertainty if you choose no, so it's just 0. Your expected value if you choose to play, is your probability of winning times the payoff of winning, plus the probability of losing, times the value of losing. If we think about that, we can think about how much value does that have to you? And that's something known as utility. Utility is basically the value of some choice to you as an individual. And, so let's say we have two possible choices, we can make, like buying a ticket, or not buying a ticket. Each possible choice has some set of consequences associated with it. Not buying a ticket only has one consequence, which is you don't win and you don't lose anything. Buying a ticket has two consequences, you win the lottery or you don't. And each consequence has some probability or chance of happening, okay.? Utility says, how much value do you get if a particular consequence happens, and how much is that worth you? And we'll talk about that explicitly in just a second. So buying the $1 ticket, in a prize with $1 million payoff, if the probability of winning, if that p is 1 over 10 million, or 1 to the 10th to the 7th power. Buying the ticket has two outcomes, win or lose. The utility of winning, assuming for right now, that you just think of the monetary value for you. Is a million- 1 in the case where you win, negative 1 in the case where you lose, and if you don't buy the ticket, it's just 0, cuz you don't win or lose. And so the expected value, basically, if you were to look at the payoff in the case where you win, times the probability. Multiply or and then look at the case of not winning versus the payoff there, the lotteries, on average, are worth about negative $0.90, okay? In California they like to have the slogan, you can't win if you don't play, and a more accurate slogan would be, basically, you can't lose if you don't play. >> [LAUGH] >> Okay. But let's bump this up a notch, okay? x is the value you derive from that particular thing, right? It doesn't have to have the actual dollar value. So, let's consider a slightly different game. Let's say I gave you this choice. I'm not really gonna give you this choice, I wish I could, but I'm not. You can choose to either play a game or not play a game. If you play the game, I will flip a fair coin, and with half likelihood, if it comes up heads, you win $20,000, otherwise you win nothing. If you choose not to play the game you just get $10,000 guaranteed. How many people would play the game? How many people would not play the game? >> [INAUDIBLE] >> We're playing it once. We play once. >> I get $10,000? >> If you say I don't wanna play, you just get $10,000. >> If I don't wanna play? >> If you don't wanna play. So how many people don't wanna play? Wow, almost everyone. But the expected value of that game should be 50% times $20,000, which is $10,000. So we would think the room would be split 50/50, why did the room all skew to one side, okay? Because the value you get from the money is actually the utility of $20,000, or the utility of $10,000, and there is not necessarily a linear relationship between those two things. So $20,000, to most people, is not equivalent to 2 times the initial $10,000. Because, that $10,000 may actually have a bigger impact on their life, than thinking about gambling with the additional 20 to get the additional 10,000 I'll make that explicit in just a second, okay? The other thing that sometimes I talk to doctors about this, is that these kind of decisions actually not just involve money, but more importantly, they involve factors like quality of life, right? And how do you trade those things off, right? There's a medicine you can take that may have a 10% chance of curing the condition and a 90% chance of being fatal. How do you weigh that over thinking about the uncertainty that there might only be a few days left, right? I don't wanna be morbid, that's a little too morbid. Let me pop-up from being morbid, but that's the kind of thing, when I talk to doctors, that they're trying to ascertain. So let's pay another game. Now this game, that if you choose to play, we will flip the fair coin, and if it comes up heads you get $1,000, if it comes up tails you get nothing. You can choose not to play, in which case you will get a guaranteed payout. And the question is, how much does that payout have to be, for you to be indifferent between playing in the game or not? So think of it this way, so indifferent, so the way you can think about indifference is, say I put some value for x like $300. You would say, you know what, to me it doesn't matter if you just gave me $300 or you entered me into this gamble for a 50% chance to win a 1000. To me that's really the same, someone else could make that decision for me and I would be fine whichever way they choose. So that's true indifference, right? You could give the decision to someone else, and you are fine with either decision, okay? So, how many people would choose to play the game if x, the guaranteed payout, was $300? Raise your hand and keep them up. >> Wait, to? >> To play the game. >> I thought you were asking. But at this point I still want to see if you want to play. So you're not indifferent. You're willing to play the game. Just take a chance. So what if I raise it to 400? How about 500? Anyone still playing at 600? There's one hand on the back. You're known as risk preferring. Right, you should go to Vegas right now because they'll give you better odds. Okay, notice there's different hands going up in different time in the room. That's not by accident and it doesn't mean that anyone is wrong. Everyone is right because to different people, this gamble is worth different amounts of money. Everyone has a different utility function. But the thing that a gamble like this allows you to ascertain is how much is the uncertainty avoidance worth to you, right? So, that's the weird thing to think about. Uncertainty itself, the probability that something will happen has monetary value to get rid of. Because this is what's known as the certain equivalent. It's the idea that if I were to have played this game, that if I were to actually choose yes to play, the expected value of the game is $500. I'm potentially willing to take less than $500 because I want it guaranteed. I want to avoid uncertainty, okay? Which tells you how much that uncertainty is actually worth to you as a monetary figure, which is a little bit weird. Okay, so the certain equivalence's potentially different for everyone. That's perfectly fine, it's what your value is. So let me give you an example. Let's say we have this choice between playing a game for $20,000 or 0, right? Flipping the fair coin or someone could say I will give you $80,000 guaranteed. And someone say was really indifferent between those two prospects. Sorry, 8,000. 8,000, that's our certain equivalent. The expected monetary value of the game is what we get when we compute that expectation thing for the game. It's just 50% times 20,000, plus 50% times 0, which gives us 10,000. That's the value of the game in expected value. But our certain equivalent is less than that. So there's actually something called the Risk Premium, it's how much we are willing to give up in terms of monetary value to avoid uncertainty. So in this case, we're willing to give up $2,000 of potential value just to avoid risk, okay? And if you're wondering, do people do this? Of course we do this all the time, it's called insurance. Okay, so let's consider a simple example of insurance. In California, you're required to buy it, in some states, you're not. But let's consider a simple example, you can choose to insure your car or not. If you insure your car, you pay $1,000 to insure it for that year. And let's just say it's totally comprehensive insurance. You get into a collision, something happens, they just pay for replacing the car, pay for any damage. Or you can choose not to pay for insurance and you're uninsured. There's a like high chance nothing happens, right? You drive along, there's no accident, there's no money you have to pay either for an insurance premium or to fix anyone's car. There is a small chance you get in an accident and let's just assume in that accident all that happens is car damage, so no one's actually injured but $30,000 of car damage. So you look at that say, what's the expected value of not buying insurance? It's negative $600, right? It's that 2% times the negative 30,000. What's the expected value of the insurance? It's less than that. We're paying the $1,000 guaranteed. Most of us choose to buy insurance. Why? Cuz we wanna avoid that risk, okay? Turns out actually something, not fully understanding this modulo, a few other things about making assumptions, about probability what many people believed was one of the causes for the housing crisis in 2000, okay? And if you were interested in that, we can talk about that offline. But I won't go into more detail. But it has to do with how we think about risk and how risk is spread out over multiple possible factors, okay? We also have a non-linear appreciation for money, which I talked about before, this utility thing. Okay, so let's play a simple, or we're not going to play it. But let me ask you for the simple game. Coin flip comes up heads, you win $10, otherwise you get nothing. Or I will give you $2 guaranteed. How many people would choose to play the game? How many people take the $2? Couple of people, very few. Now I'm gonna take those exact same figures and make a linear transformation, which just means I multiply them all by some value. I'm gonna multiply them all by the value of 10 million. >> [LAUGH] >> Okay, so you can take the coin flip for 100 million or nothing, or you can take $20 million guaranteed. How many people wanna take the coin flip now? >> [LAUGH] >> One person, all right, we're going to Vegas. We're. >> [LAUGH] >> And the reason is is that the amount of change in someone life that happens by getting $20 million and having certainty around that is worth much more than the potential for another 80 million, all right? Event hough the expected value of this game. Sorry about that. The expected value of that higher game is worth more. It's because that extra 80 million won't significantly change our life in a way that the first 20 million won't have already done. And get rid of a bunch of worries for us. And that's one of the things we wanna think about is that when we make these kind of decisions, thinking about the non-linearity of how we value these things makes a huge difference. There's these things called utility curves and risk profiles, and the basic idea is if you're risk neutral, which very few people actually are, the first $50 of, I'd say think of this graph as being utility versus dollars. The value of the first $50, the size of that bar as the same as the next $50. Most people are not like that, most people are Risk Averse. The second $50, so going from 50 to a $100, has less value to them than the first $50. And I feels like it's hard to imagine this with $50, make it 50,000. Make it 500,000, okay? And then it becomes more clear for each individual, depending on their utility curve, why the next increment of the same amount of money is not worth the same as the first amount. There are some individuals that are Risk Preferring. They tend to really like gambling cuz the second $50 is worth more to them than the first $50. And so if they can take an even, fair bet that's 50 50 to win that second $50, they're likely to do it. But there's very few people like that in the world. I shouldn't say very few people, there is a small number, okay? So, I need another volunteer. Come on down. So, here's Thomas Bayes asking for another volunteer. So, what we are gonna do is we're gonna play a slightly different game than what we had before. So before we had three envelopes. This time, I'm gonna have two envelopes and I'll allow you to have one. I'll tell you the set up for the game. One of these envelopes contains $X. We don't know what X is but one of the envelopes contains $X, the other contains $2X. So one has double the amounts as the other envelope. Pick an envelope. All right, don't open it yet. So you have your envelope, okay? So you selected an envelope. I will give you an opportunity to switch. Do you wanna switch? Does it make any sense to switch? No. At this point, we have complete uncertainty. You have 50% chance of getting either one of the envelopes, 50% chance of the other envelope, okay? Now, go ahead and open the envelope. >> [LAUGH] >> So open up the green paper, that's so you can't hold it up to the light. What's in there? >> $20 bill. >> $20 bill. Now, would you like to switch? >> [LAUGH] >> So let me help you out with this, right, cuz it's sorta like I'm putting you on the spot. And i know it's kind of rough, it's a Friday afternoon, you're in front of like 250 people, let me help you out. Okay, so let's say y is the amount of money in the envelope you picked, right, so we're just gonna call that y. So how much, what would we expect is in the other envelope, right? So what you have y. What we expect in the other envelope is one half chance that there's y over 2, right? >> 45. >> Well, so let's just do- >> I mean 50, sorry. >> Yeah, so we'll go one step at a time. So you're ahead of me, that's really good. But I'm gonna go a little bit slower cuz I move a little slower. So I have 50% chance the other envelope has half as much. And I have a 50% chance the other envelope has twice as much. So when I multiply these together, this gives me a y, and this gives me a one-fourth y. One-fourth y, which gives me five-fourths y. Which is strictly greater than y, for any positive value of y, right? Which means, if you have $20, the other envelope is worth $25. Should you switch? >> Depends, am I risk-averse, risk-neutral, or risk-liking? >> Yeah >> I would say, how about, no. >> You could say no, that's perfectly fine. Okay, let's say you were risk-neutral. So let's say, that actually all dollar values, at least in this range, so there is some small range of dollar values that people are actually risk-neutral in. So let's say you're risk-neutral. Would you wanna switch? >> Yes. >> Okay, now here's the question. That envelope, before he opened it, I could have written y on front of it, and that would have been this y. And this envelope would suddenly become worth five-fourths y. So before even revealing how much is in the envelope, as soon as you picked an envelope, I could make the claim that this envelope is worth more. But before we opened anything, it was like, it's not worth it to switch. So what happened? Okay, I got really disturbed, that's what happened. >> [LAUGH] >> All right, this is one of those things where, when we think about the probabilities, risk aversion, you're absolutely right. It's first thing we wanna think about, in terms of potential payoff we get. The other thing we want to think about is, what is a probability, okay? Which is a very, it turns out, interestingly enough, when we talk about probability. People talk about, they're like, it's a mathematical concept, we should know what probability is. There are two different schools of thought around what a probability actually means, in some semantic sense, which is weird, when I tell my students that, they freak out. They're like, it's math Marin, it's numbers, what do you mean there is different interpretations. One of the interpretations of probability is basically just, you see something happen over and over, and that's how you determine the probability. I roll this die a million times, figure out what fraction of the times I rolled a one, and that's the probability of getting a one. If I were to roll the die infinitely often, that's called frequent disprobabilty. There's another form of probability called Basesan probability, which is named after our little friend Thomas Bays, that says all probability is, is a belief. And if you think about that, I can ask you, what's the chance it's gonna rain tomorrow? Most people would say, maybe, I don't know, 10%? Have you seen infinitely many tomorrows to make that determination? No, we haven't seen any of them. But how do we make that ascription? Well, we've seen the general weather pattern on that particular day for many years in the past. But it's not the actual day tomorrow. So, in one level of interpretation, a probability is just a belief, and part of that belief comes down to, what values do you think I would put in the envelope, right? I'll would bet you would be shocked if you open one of the envelopes, and you're like, you put a million dollars in this envelope. I don't care about switching, I'm just leaving, can I keep it? >> [LAUGH] >> Right, but that's probably not something you believed beforehand. And so the issue with this, before you open the envelopes, is either one is actually equally good, okay? But what happens when we actually open the envelope and put the value down, is this little equation I gave you here, is actually not true, okay? Despite the fact that it looks true, and it's easy to sorta take you through that exercise. Here's is what's actually is going on. So the setup again, you know there's one that has x dollars, one that has 2x dollars, okay? And I give you this garden path equation, because it's leading you to the wrong place. The problem would actually happen when you open the envelope. Is for this equation to be true, you have to believe that the other envelope really is equally likely to have twice as much money or half as much money, no matter how much is in the envelope you pick, okay? And so if you pick an envelope that had a million dollars, you're like, man, either Stanford's really paying well these days, or this guy's probably not gonna put 2 million in the other envelope. It's probably more likely that he would put half a million. [LAUGH] Probability is 0 I'm putting either of those amounts in there. >> [LAUGH] >> But mathematically, what it actually means is, if you assume that any possibility of twice and half was equally likely, you would basically be saying on the number line, there is equal probabilities for all values from 0 to infinity. And I need to be able to integrate that, and have it integrate to 1, because all the probabilities have to sum to 1. There's no constant value, just to bring back shades of calculus. There's no constant value greater than 0, that I can integrate from 0 to infinity that equals 1. So this thing can't exist, this equation can't exist. So the only thing that can exist is, we have to assign for ourselves what those probabilities are, okay? So, if that's the case, then we have this subjective belief about what's in the envelope, right? So we can't say all values are equally likely, that's not a true probability distribution. And so the frequentist, that interpretation says, well you play the game infinitely often, and see what values Marin puts in the envelope. I'm not going to play the game infinitely often [LAUGH], cuz I just lose every time. >> [LAUGH]. >> That's the problem, is I only let you play once. What the Bayesian says is, I have a prior belief. A priori I believe, you know what, there is probably some distribution where I saw him do the 20 with the Monte Hall game before. Maybe the chance that he puts a $20 bill in there is like, I don't know, 20%. And so the chance of, that he's actually gonna put $40 in there, I might say, is actually only like 5%. So the chance he might put $10 in there is probably like 50%, and so you come up with some distribution. And once you have that distribution, what actually happens when you open the envelope is, you see where on that distribution you are. And now, given the probabilities that you have, which might also get updated slightly, because when you see what value it is, you think, he's willing to give more or less money. Then you can do the math of multiplying the probability times the payoff in choosing the other envelope. So with all that said, do you want to switch or keep it. >> [LAUGH] >> I'm gonna stay with, I might keep, I'm keeping it. >> You're gonna keep the 20, awesome. All right, so I could tell you what was in the other envelope. >> Was it a 20 to make the story worth it? >> No actually, you made a good choice. Cuz the other envelope has $10 in it. >> [APPLAUSE] >> Well done. >> [APPLAUSE] >> You can keep the 20, but i would like the envelope and the green paper back. >> [LAUGH] >> Thank you. So, here is another way to think about it. What the Bayesian would say is, I have this distribution over the values. And I'll make it really concrete for you, okay? What they would do is, when they have the distribution over values they would say, once I open the envelope, I can now think, what was my expected value, based on my distribution, versus what I saw? So, is it actually more likely that there's $10 in there versus $40, multiplied by the probability to get the expected value? Here is an easy way to think about it. What if you opened the envelope, and instead of $20 in there, there was $20.01. >> [LAUGH] >> Would you switch or not? >> Switch. >> Yes, >> Why would you switch? >> [INAUDIBLE] >> The dreaded half cent. What about the half cent? And you're like, but actually Marin, half cents were stop being minted in 1865. And if I got one in good condition, it'd be worth about $150. So I'm switching no mater what. Notice the amount of, without even knowing about half cents, the amount of information you bring to bear in that problem without thinking about it, right? So if there's $20.01, you say, I should switch because one, there isn't gonna be half cents, or if there are, maybe they would actually be worth more. But i never told you that the dollar values in there couldn't be some fraction. I never told you that it had to be US currency until you actually opened the envelope. I never told you that these weren't possibilities, that there had to be interval quantities. And there, some people say, but you wouldn't put change in there. Like if it was $20.02, you wouldn't put change cuz, then I could shake it. I never told you any of those things. All of those assumptions were part of your decision making process that led to the decision that said, if there's $20 in one sense I should switch. And that's the other thing about decision making, is not only the probabilities, but understanding what are the implicit assumptions we actually use to make those decisions when we think about the set of options that are actually available to us. Which could be larger than we otherwise think. Okay? So, let me give you one last problem. So, just to in your Friday afternoon, you might leave a little disturb to wonder how rational you actually are. Okay? So, I'll give you 2 different choices and you can pick 1 option for each choice. So, choice number 1, you have 2 options, A and B. So, option A, you got a million dollars guarantee. Option B there is an 89% chance you get a million dollars, a 1% chance you get nothing and 10% chance you get $5 million. So go ahead and choose. Don't say that aloud, just choose that in your head which one you would choose. Okay? Option 2, not so nice not guaranteed options. So you have a choice for C and D. Where C says there's an 89% chance of nothing and an 11% chance of $1,000,000. And D says a 90% chance of nothing and a 10% chance of $5,000,000. So go ahead and decide for yourself which one of those 2 you would choose. Okay, how many people chose A and C? B and C? [INAUDIBLE] >> Okay go ahead. >> [LAUGH] >> I know is a tough decision. >> [LAUGH] >> Okay. [COUGH] [COUGH] >> You got your choices? Got them? All right. Anyone choose A and C? Few folks. Anyone choose B and C? How about B and D? Fair number. How about A and D? All right so we got kinda a split between B and D, and A and D. Interestingly enough if you chose A and D. You are irrational. >> [LAUGH] >> For any choice of utility function. Doesn't even matter how you value money as long as you value it as non-zero. >> [LAUGH] >> Why? So let me show you mathematically why this is a case and then I'll show intuitively why psychologist say this is the case. So mathematically, if we prefer choice A to choice B, then we would prefer 100% chance of the utility of $1 million would be greater than 89% of the utility of a million plus 1% of the utility of 0 plus 10% of the utility of 5 million. So now I'm gonna do, is look at that same thing for choice D. If I make choice D, what I'm saying is that the utility of 89% of the utility of 0 plus 11% of the utility of a million is less, cuz I didn't choose C then 90% of utility of zero plus 10% of the utility of 5 million. Now I'm gonna do a little Math. So I start of with choice D and I'll say hey, I have a little equality subtract 89% of the utility of 0 from both side. Okay, so I'll eliminate the 89% of utility of the 0 on top, on the bottom I'll have now 1% of utility of 0. Then I say go ahead and add 89% of the utility of $1 million to both sides. So I do that and so the 11% of the utility a million becomes 100% of the utility of a million. And I also add an 89% to the utility of a million. And I have the boxes for choice D up there just rewritten. And choice A that looks super similar. Except for the fact that one is greater than and the other is less than, okay? So the choices are inconsistent, okay? And that's for any choice utility the function cuz I didn't tell you what the utility values were I just did it at the function of utility. Why is that? Why do we make those decisions? Okay, so there is this thing called 'Lay Paradox'. Thanks to Mary Suley, who is a French economist, who actually won the Noble Prize. But the reason for this, is that essentially what this boils down to, is we value certain D more than its worth. Okay. So in the first choice, when there is a 100% chance of A, when we choose A, we say that's a million dollars and I know there's this 1% chance I might lose in there. There's even a 10% chance I might get 5 times as much. But if I were to lose I'd have so much regret and I don't wanna do it that I want the guaranteed win. Well guess what's going on with the choice between C and D. In order to get the 5 times payoff, you are giving up a 1% chance of winning anything. So it's that same 1% chance which makes it 1% more likely you will get 0 in the case for D. That's exactly the same 1% as getting that 1% in the first case of getting 0. But in the first case the situation is framed as certainty verses uncertainty and in the second case both decisions have uncertainty associated with them and so we don't value you them the same even though they should be by some normative rule. Okay. And that's what psychologists say will happen. They will say that we tend to over value certainty in life. When it actually exists. So to wrap up, I would just say decision making often occurs under uncertainty. There's these probabilities out there. But if we can somehow understand the decisions that we're making from the standpoint of understanding probability. And the expected value we get, and the hidden assumptions we make when we make those decisions, and what's really important to us at the end of the day, hopefully we can make better decisions. And remember, not all decisions are subject to decision analysis. So thanks very much for coming. >> [APPLAUSE]

Contents

Explanation

Some form of win probability has been around for about 40 years; however, until computer use became widespread, win probability added was often difficult to derive, or imprecise. With the aid of Retrosheet, however, win probability added has become substantially easier to calculate. The win probability for a specific situation in baseball (including the inning, number of outs, men on base, and score) is obtained by first finding all the teams that have encountered this situation. Then the winning percentage of these teams in these situations is found. This probability figure is then adjusted for home-field advantage. Thus win probability added is the difference between the win probability when the player came to bat and the win probability when the play ended.

Win probability and win shares

Some people confuse win probability added with win shares,[citation needed] since both are baseball statistics that attempt to measure a player's win contribution. However, they are quite different. In win shares, a player with 0 win shares has contributed nothing to his team; in win probability added, a player with 0 win probability added points is average. Also, win shares would give the same amount of credit to a player if he hit a lead-off solo home run as if he hit a walk-off solo home run; WPA, however, would give vastly more credit to the player who hit the walk-off homer.

Baseball

MLB postseason

In Game 6 of the 2011 World Series, St. Louis Cardinals' third-baseman David Freese posted the best WPA in Major League Baseball postseason history, with a 0.969, which was 0.099 better than the now-second-best WPA of .870, posted by the Los Angeles Dodgers' Kirk Gibson in Game 1 of the 1988 World Series. The third- and fourth-best WPAs are .854 (by the San Diego Padres' Steve Garvey in Game 4 of the 1984 National League Championship Series) and 0.832 (by the Cardinals' Lance Berkman in Game 6 of the 2011 World Series).[3]

References

  1. ^ Keating, Peter (2011). "The Next Great Stat". ESPN The Magazine. ESPN. 14 (1): 116–118.
  2. ^ http://www.advancednflstats.com/2008/08/win-probability.html
  3. ^ "David Freese: now THAT was the best World Series performance in history". Baseball-Reference.com. Sports Reference LLC. October 28, 2011. Retrieved 2011-10-30.
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