確率とオッズ：ヴィグ（胴元の取り分）って何？

Welcome. My name is Patrick Curran, and along with my highly probable friend Greg Hancock, we make up Quantitude. We are a podcast dedicated to all things quantitative, ranging from the relevant to the completely irrelevant. In this week's episode, Greg and I explore the fun, but sometimes confusing world of probability, odds, log odds, and odds ratios, both when encountered in statistics and in everyday life. Along the way, we also discuss empty airplane hangers, drink carts, kiaoshi, Costco hot dogs, the Tush Push, Quantitude International, the Blues Brothers, spelled with a Q, the Vigorous, Mr. Peabody, a probability of 1.8, what the log giveth, and horse steroids. We hope you enjoy this week's episode. Dude, we're starting. What are you doing? Just give me a minute. You are so impatient. I get like one phone call a month, and I got a voicemail from Raleigh-Durham International Airport.

So hang on a second. Okay. All right. This message is for Mr. Curran. I just wanted to let you know that we have confirmed your extended rental of one private airplane hangar here at Raleigh-Durham International Airport and have enrolled you in auto pay with the first three months prepaid. Please contact us when you are ready to move your jet into the hangar or if you have any questions at all. We look forward to working with you and your whole team at Quantitude. Oh, dude, I told you during the episode that we had not thought all the way through how to monetize through crime. Yeah. In fact, I've been going through some of the Quantitude financial statements lately, and it's, well, yeah, it's not great. It looks actually like we would not be able to afford a jet for, I think the projection was 1711 years. I mean, it's plus or minus, right? How does that work, though? Because we literally do not have an income stream. So I'm a little curious what algorithm you're using.

I don't know. The projections on Claude seem pretty solid. So there's a little good news. I got on eBay and I bought us a used beverage cart from an old DC 10. So that's good. That's cool. If it's a private jet, isn't there somebody who brings you stuff? I don't think they have a beverage cart. Okay. Now, what we could do is next time I come up, we could put it in your minivan and ask Goldie to push it up and down and offer us rent. I think that's going to go super well. That's what it is. We need a revenue stream. I want that jet. I know. This American life has a jet, and I want one. I know. So I was looking at my phone the other day, and I keep getting ads for this. Is it pronounced Calchi? Have you heard of that before? The online betting futures thing. Yes. It is so cool because you can bet on anything. You can bet on the weather.

You can bet on whether a sports announcer is going to use the phrase tush-push football. Thank you for clarifying. You can even bet on whether or not Costco is going to change the price of their hot dog and a beverage. Costco, the $1.50 hot dog and soda combo, unchanged for nearly 40 years. You can bet on stuff like that. You can also bet on what Trump is going to say in his speech, but when you're the speech writer and put the thing in that you bet on... The White House employee allegedly used his advanced knowledge to place bets on the words the president would use in speeches and make money. Okay, so how about this? Yeah. What do you think of Kalshi? I don't know what you just did with your mouth. Well, let's try it. It's a little harder to say. Kalshi. Kalshi. So it's like a Minnesota version? Kial. Kial. Kial. Kial-she. Kial-she. Kial-she spelled with a Q.

Okay. Work with me, man. Q as in copyright infringement. No, no. Our lawyers at Quantitude International said that if we try to pronounce the Q, Kial-she can't sue us. Okay. So I propose we launch Kalshi. Kalshi. Where you can bet on futures of stupid stuff that we say on the podcast. Wait, we say stupid stuff? No. Okay. I just won $1.50. What stupid stuff do I say? What would you bet on that I'm going to do or say in our episode? Let's go. I'm looking at you. All right. I will bet $5. that you say the same phrase twice in the same sentence. So you will say, when we compare models, compare models. I'm going to put $5 on that. I do not do that.

All right, all right. So I would bet on whether or not you make a pre-1990s movie reference bonus points for the Blues Brothers. That's a great movie. It's 106 miles to Chicago. We got a full tank of gas, half a pack of cigarettes, it's dark, and we're wearing sunglasses. Hit it. Okay, so I made a new account, alright, already, at Kalshi.com. Alright. Maybe we can just say Kalshi and then go ahead in your radio voice. With a Q. There, okay, so I made an online account at Kalshi. With a Q. I put $5 on that in this episode you say, I don't know if I've mentioned this before, but I went to a drama school. Come on, we had seven productions a year, two musicals fully orchestrated, Shakespeare, dinner theater, play festival. We had a large stage. We had a small stage. Whatever. Jealous. Okay, okay.

One more for you. Oh, no. I'm going to lay down $10 that at some point in this recording you are going to say it was the best of times. it was the worst of times. Come on. Even Kai Risdahl invoked that a couple of days ago. With apologies to Charles Dickens, it is the best of farm economies. It is the worst of farm economies. I love you, brother, but that's going to happen. It's a solid bet. Yeah. What do we have to do to get this... How was it? Kia? Come on, man. It's like you're not even trying. Okay, let me just channel some of the training I had when I went to a drama school. Woo-hoo! Damn it! All right. Anyway, once we get this launched, how are we going to get people to use it? Oh, see, this is a good question that we should have considered with the true crime one. All right. Now, the usual evil online betting places push credit to people's accounts. Oh, right. They give you $250 in free money to bet, and then you lose it, but you've already made your account, and so you keep betting.

Wow. Oh, this is like the Dark Lord runs some of these online things. If you only knew the power of the dark side. I don't want to do that. Okay. You know what I think we should do is encourage people to unearth their memory of probability and odds and betting. Ah. So that they can participate in Kyal Shih. With a Q. There we go. Not going to be annoying at all. because this is stuff that we've all learned about before, thought about before, but that we need to be reminded of this. And also it going to let me use one of my favorite words in the English language Later in our discussion I going to be able to introduce the VIG Ooh that quite a tease So let me start with just from my limited exposure to the original Kalshi. I know that everything that you bet on has what they call a yes price. And it's some amount that goes from zero cents all the way up to 100 cents.

And they assign a yes price of like 80 cents if Patrick mentions the Simpsons or whatever it might be, which is effectively the way that they are trying to communicate information about probability. So why don't we start off with probability and go from there? This one should be fairly quick. All of us are familiar with probability to varying degrees. We've been taught about it in school, but there's also a lovely common sense element to it. A probability is a simple ratio. There's a numerator and there's a denominator. That ratio is bounded between zero and one. In the numerator is the number of ways a particular event could occur. So imagine you have a single die. What is the number of ways that you could roll a two? Well, there's only one way that that can happen, and so that goes in the numerator. What about the number of ways to roll an even number? Well, there are three ways because you can get a two, a four, or a six. Three would go in the numerator. So however many ways the event of interest can happen goes in the numerator.

And then what goes in the denominator is what we sometimes call the sample space. And that is all the things that could happen in that experiment. All right? So that is the sample space. What are the total number of things that could happen in rolling the die? Well, there's six possible things, one through six. If you're interested in, well, what is the probability of rolling an even number on a single die? Three goes in the numerator. Six goes in the denominator. It's one half. You have a 50-50 chance. That's all probability is. A couple of things I might add to that. One is, in your description of the sample space, when you roll a die, each of those outcomes, one through six, they're equally likely. If they weren't equally likely, then we could apply some kinds of weights to how likely those are. And your definition of probability applies to so many things. There are some things that it's harder to assign a probability to. If someone said that they wanted to assign a probability to the chance that a football announcer is going to use the phrase tush-push,

well, what they could do is they could go back and look at all the broadcasts and figure out what proportion of them was that mentioned, and then they could set a probability based on that prior information. If I am betting on whether or not Costco is going to change the price of their hot dog, something that they haven't done, that's a much more difficult thing to set some kind of probability to because I can't even envision what the whole sample space is. So there are different ways of getting probabilities. Once we have that, then we are ready to go. And what we really want to focus on here is that just super simple, If you have a probability of 0.75, there is three in the numerator, there's four in the denominator, there are four possible outcomes, three of them are the event, and there's a 0.75 probability that that event is going to occur. Now, can we go back to the mid-1600s? Where are we going today, Mr. Peabody? Not where, Sherman. When?

What we want to do is move from probability to what are called odds. And I've got to tell you, there's a very, very simple relation between probability and odds, but a great deal of confusion about how do you interpret those in practice? Why do they exist? We can go back to the 1600s, and there are a ton of people involved, but a really famous interaction was between Pascal and Fermat, and they were dealing with this thing called problem of points. And it was a thought experiment. It was a really interesting math-based probability thought experiment, which is what if there's a gambling game that is happening? There's a pot. You've been playing it, but the game ends before the logical stop of the game. So the game ends early, and they were interested in what is a fair distribution of the pot given that point in the game. And it was really just a math problem. It was a probability problem.

Given what's happened, the events of what happened, who is bet, what is the money in the pot, how would you give everybody their money in a fair way? That's kind of seen as one of the really important first steps forward in moving from probability to odds to understand wagers in gambling. The thing that I love about the history, by the way, is that we often think about probability as being so firmly rooted in mathematics. And of course, there were people who were very good mathematicians working on it. But so much of this really just came out of gambling. So you said about converting probabilities to odds. If we frame this in its historical gambling context, probability is going to tell us something about how likely a particular payoff is going to be. But when we place a bet, we have to connect that likelihood to what we risk and how much we would actually receive. So odds is just going to be the probability of an event in the numerator divided by the probability of not that event, or 1 minus the probability.

So that means that if the probability of an event is 0.20, probability of that event happening, then the probability of that event not happening is 0.80. 0.2 over 0.8 is a ratio of 1 to 4. And we will often refer to that as the odds, 1 to 4 odds. One success for every four failures. And that is where I think there's a lot of confusion. If some of you are listening and saying, no, no, no, it's four to one. Because often in gambling, you put the larger number first. And it all depends on is it the odds of success or is it the odds of failure? Yeah. And when people have a convention of putting the larger number first, four to one, you have to be very careful that you know what those odds are in favor of. May the odds be ever in your favor. Actually, in some of the gambling apps, it's intentionally confusing, and that can be a very helpful thing from the point of view of making apps money,

but not so helpful or fair to the person who's placing the wager. So how do I actually translate then from these odds into a wager that someone would make? This is all based on a thing called a fair wager, which is if you have uneven likelihood of an event occurring, you want to be able to make a bet that is proportional to the risk that you're taking. So let's say that you have three to one odds of a successful event occurring. Now, imagine that you want to bet on the failure. All right. So you've got a three to one odds and you want a fair bet on the failure. The three to one literally means that if you bet a dollar and win, you win $3 in profit. That is, you get $3 for every $1 you bet. So you get your dollar back. Yeah, you get your dollar back plus $3. Now, one might ask, did the bookie who's running the numbers make a similar error that we did in the true crime episode without monetizing it?

Which is, if you do this in this way, you're never going to make money. Right. You would just break even every time right You would break even every time Okay And for you statisticians at home this is where expected value came from What is the long run expected value of your winnings in a given situation, right? This is where you get to use the VIG, right? You love saying that. I can tell that you love saying that. Well, it's in like the parlance of language, but I think the history isn't always respected. Somebody will say, you know, coming out of a faculty meeting, I've actually heard somebody say, after a dean presented, it's like, well, that sounds pretty good, but I wonder what the vig is. And I kind of want to say, first, it's cool as hell. It is cool. The dean is pitching a thing that seems too good to be true. And you've got to ask yourself, what's the vig? The word of the day is vig. So you have Yiddish background in your blood.

Tell us what a vig is. You're saying this like it was common in my family that my grandmother would go, Gregory, what's the vig here? So, yeah, it's slang for winnings or profit, the vigorous. It is a cool thing. I don't tend to work that into my common everyday language, but I just feel like I would sound so much cooler if I did. Right, right. So what's the vig? The vig? Yeah, man. What's the vig on this action here? Do you even know what vig means? So what that is is the bookie who's running numbers or Calci behind the scenes, the VIG is there might be a 10% cost of placing the bet. So if you bet that dollar and you win the $3 in profit, you lose $0.30 out of that for the VIG. That's what the bookie is winning. But what this takes us to then is we have a 3-to-1 odds. We have an 8-to-1 odds. But in wagering, it doesn't always have to be to the one.

So you could have a five to two odds. Okay. Why would you have a five to two odds? Well, it keeps you in integers. All right. If you bet $2, you win $5 in profit minus the VIG. We always got to remember that. It's $5 in profit minus the VIG. Let's back calculate what that means. If we've got a 5 to 2 odds, there are five ways of success. There's seven total ways, and that's a probability of 0.714. Probability of success. Probability of success. So if you've got 5 to 2 odds, that's actually a probability of 0.714. If you have probability of 0.714, that's an odds of 5 to 2. They are the same thing. They just convey different information. Yeah, the odds are really, really useful if you think in terms of payout, right, which is what the purpose and the origins of this are. We as statisticians might have other reasons for using odds, but that doesn't involve getting to say the VIG.

So this is very simple. Exactly, exactly. We need to work that into like logistic regression. But this point is really an important one. Everything that we've talked about is just we're trying to get a jet to take us somewhere with a drink cart. So you're exactly right, as working with integers in the odds, having a 5 to 2 helps us understand the risk and the payout for a success and a failure in rounded dollar amounts. But when we move to statistical analysis of odds, we actually don't have any requirement to use those integer values. And so when we calculate a p divided by 1 minus p, we can just have a single numerical value that's not an integer and work with those in the usual way. And it turns out there are a bunch of advantages to that. Yeah, there are advantages, maybe not as many advantages as there could be, but there are advantages.

And what I mean by that is, imagine we want to predict the probability of a particular event. If I'm using a linear model for prediction, then what's going to happen is that I might predict that the chance that Patrick mentions, the Blues Brothers, is a probability of 1.8, right? I can get predicted probabilities that go out beyond the upper level, or I could predict a probability that might be minus 0.2. And those things, of course, don't make sense. One of the nice things about odds from a statistical standpoint is that whereas probabilities go from 0 to 1, odds go from 0 all the way up to positive infinity. So what we're really doing is we're taking the ceiling off that so that the odds can go up, up, up. and that gives us so much more room for prediction. And again, it keeps us in that space where we can think about the ratio of an event happening versus not happening. Okay, so here's an interesting thing. We have a probability that I was talking about before, a 0.714 with a whole lot of little digits after it,

which is 5 divided by 7. That is the 5 to 2 odds. But what we can also do is when we calculate the odds from that probability of 0.714 with lots of little digits, it's 2.5. So there are a couple of interesting things to see. One is when we start using these in statistics, there's absolutely no reason that we have to use integers. And so the odds are 2.5. And indeed, you may well have odds of 8.632, 5444. There's absolutely no reason these have to be rounded. The only reason that historically we use them as rounded integers is it's for risk and payoff in bets. That's all. Five to two is an odds of 2.5. Now, we literally can say you are two and a half times more likely to experience the success than the failure. That is how you can interpret that odds of 2.5. All right. So, cool. Now, what Greg did when we moved to the odds is we took off the upper limit of the 0, 1 probability.

Odds are bounded between 0 and positive infinity. But we still want to fit our linear models to that, and that's an asymmetric relation. We're still bounded at the bottom, and we don't like to be bounded. So what we do is shrug and say, well, if we take the natural log of the odds, then we take off the bottom limit, which allows us to go to negative infinity, and it runs to positive infinity. So the log of the odds undoes the top and bottom, and that's what we call the logit. That's so beautiful. You've got this middle point that is zero, and it's perfectly symmetric with all the positive stuff and the negative stuff. It is why I like my dishwasher loaded so neatly. The logit is this beautiful symmetric thing, and now we can start fitting linear models and not worry about making predictions out of bounds. We know exactly what zero means. A zero in this logit or log odds scale means the same thing as odds of one to one, means the same thing as probability of 0.5.

It is the harmony of the spheres for us statistically. And that is literally logistic regression. We fit a linear regression model to the logits, the log of the odds, and then go ahead and give it horse steroids, and it moves into nonlinear SEMs. It moves into everything that we do with discrete outcomes. We can have odds. We can have proportional odds. We can do all sorts of wild stuff. But this is where it all comes from. It starts with the probability of an event. and then we take the probability of the event and we divide by the probability of not the event, and then we take the natural log of that, and then we fit our linear model to that logit. But here one of the beauties is what the natural log giveth the exponent taketh away Yeah as beautiful as fitting a linear model to these logits is when I say a one unit increase in my predictor yields that particular slope change in

the logit, I don't speak logit. So to be able to pull this back to something that I really understand, especially as a gambler, I would understand, when we exponentiate that slope that we have in the linear model, it brings us back to something that is in the odds metric that I have a prayer of actually interpreting. And that's the odds ratio. It can either be a categorical comparison. You look at the odds in group one versus group two. If you have a continuous predictor, you can say a one unit change on the predictor is associated with this unit change in the odds, in that odds ratio. But here's the tricky part. Remember that I said that if we have a five to two odds, and that's an odds of 2.5, and we can say that the success is two and a half times more likely than the not success for just the odds. We can't say that anymore when we move to the odds ratio.

And the reason is that probability is bounded between 0 and 1, and it has this beautiful S-shaped curve, and the amount of implied change in the probability depends where you're moving on that curve. Because if you're down low on that curve, there's a small amount of movement. If you're up higher on that curve, there's a greater amount of movement. So what that means is if you get an odds ratio of, say, three, this is insanely important. You cannot say that the event is three times more likely in group two than it is in group one. And that's what we all want to say. All we can actually say is the odds is three times larger in group two than it is in group one. And we got to do a little bit more work to compare the probabilities. Think about having a continuous X, a single continuous predictor.

And in that beautiful S shape that you described, and let's assume that that S is in the positive orientation, that when you are down there at really, really low probability events, you might go from a probability of something occurring of 0.01 to the probability of that event becoming 0.02. You just doubled the probability, right? But if the probability of an event is 0.4, what would you have to do to double the probability? You would have to get it all the way up to 0.8. And so these probabilities don't function the same way throughout. And I think that sort of gets lost in some of the things that we do. We can address it, though, by calculating a thing called a risk ratio. We don't see this a lot in practices. I think a lot of times you fit the logistic regression, you get the odds ratio. You talk about that the odds are three times higher in group two than group one.

But with a little bit of extra work, you can actually do what Greg just described. And wherever you're falling on that beautiful S curve, you can get what is actually the probability in group two. What is actually the probability in group one? And just take the ratio of those. Yeah. So it's certainly a calculable value. It's just we need to rearrange our information a little bit to determine where on that S-curve you're moving one unit. I do not tend to teach these risk ratio kinds of things, but I think they are really useful to ground people's understanding when they're talking about probabilities and odds. And I think I should do it more when I'm teaching, for sure. All right, so how can we monetize all of this? Kialshi. With a Q. Yeah, but now I'm yodeling somehow. Kialshi. Okay, we're going to focus group for this and we'll get back to you. But let's go back to first principles and bring us through to what we've talked about.

We start out with a crazy simple notion of what is the probability of an event occurring, however you define that. And we would like to know what are fair bets. What's a fair wager? What am I going to risk? What am I going to win? And so the odds are P divided by 1 minus P. So we can say 3 to 1, 4 to 1, 5 to 2. We can divide those out and instead of 5 to 2 have an odds of 2.5. And you're two and a half times more likely for that success to occur than not. And then we take the log of that as the logit. We fit our linear model to that and do everything in the usual way that we do. And then we back out all the way back to the probabilities. It is absolutely gorgeous. It's like the prodigal son, right? I went to Sunday school, is we start with the probability and we go out in the world and then we come all the way back to the probability and your father kills you a fatted calf in celebration of your return.

All right. Who out there bet on Patrick mentioning Sunday school? Anyone? Anyone? Who had that? Okay. So we have you all set up for you to give us your money. And if you can kind of get on this, we have an empty airline hangar at Raleigh-Durham International Airport. Throw us a bone, man. We need your money. All right. Thanks, everyone. Take care. Bye-bye. Thanks so much for joining us. Today's episode has been sponsored by, what else? Kelshi with a Q. Now you can place bets not just on what we say on Quantitude, but also about academia. Today's even money bets. Will a student email, is there anything I can do to improve my grade? Within six minutes of final grades being posted? Odds one to one. Will an administrator describe a budget cut as an exciting opportunity? Odds one to one. Somewhat riskier options. Will an entire Zoom meeting take place where no one forgets to turn on their microphone?

Odds five to one. Will the quick scheduling poll find a time next week when six faculty members are all available to meet? Odds, 9 to 1. And for those who like to keep things really spicy, will reviewer 2 refrain from recommending a paper by reviewer 2? Odds, 20 to 1. And will the person who says in a meeting, I'll keep it brief, actually keep it brief? Odds, 40 to 1. And finally, today's Kjalshi Longshot of the Day at 1,000 to 1 odds, Will the dean conduct a faculty meeting without saying any of the following? Thought leadership, reimagining, robust dialogue, moving the needle, capacity building, growth mindset, resilience, transparency, breaking down silos, future ready, impact, transformative, leveraging resources, strategic alignment, student centered, shared governance, synergy and stakeholder engagement. Win that and you can fund your own endowed chair. Be sure to tell your friends to subscribe to Quantitude on Apple Podcasts, Spotify, or wherever they go for things with relatively good odds of hearing,

It was the best of times. It was the worst of times. You can also follow us on X, Blue Sky, or Instagram, where we are at QuantitudePod, and visit our website, Quantitude.org, where you can leave us a message, find organized playlists, show notes, syllabi, and other fun stuff. And finally, get cool Quantitude merch like T-shirts, hoodies, mugs, stickers, and spiral notebooks from redbubble.com, where all proceeds from non-bootleg authorized Quantitude Pod merch go to donorschoose.org to help support low-income schools. You have been listening to Quantitude, the podcast whose VIG has been higher than its payoff since 2019. So what's the VIG?