Las Vegas knows people love to bet and Las Vegas loves people to bet. They can set bets that are favorable to Las Vegas. So in the Super Bowl or just about any sporting event, there'll be a lot of crazy bets and these are called props bets. And so if you know a bit of basic math you can figure out if the props bets are in your favor or not. For instance, you may remember in the 2013 Super Bowl, the first score was a safety. You knew Denver was in trouble right then. Or 2014 Super Bowl during the 2013 season. You knew Denver was in trouble when the first score was a safety, so you could bet on whether the first score in a Super Bowl is a safety, a touchdown, a field goal, etc. And basically we'll talk about that a couple of videos from now. But basically you can bet how many interceptions Tom Brady's going to throw in the Super Bowl. You can bet basically will Peyton Manning throw a pick 6. You can bet on just about anything. So let's talk a little bit in the next few videos about how you, if you were Las Vegas, would set the odds on these crazy props bets. Here's one of the craziest ones I've ever seen. When the Ravens were in the Super Bowl, Shaquille O'Neal was busy playing in the NBA doing a great job except he missed a lot of free throws. Okay. So here was the bit. You take Dilfer's completions in the Super Bowl minus Shaquille O'Neal's missed free throws they were playing a game that day. If that was greater than four, a bet on Dilfer wins, if that was equal four it was a tie or a push, nobody wins. Otherwise the Shaq bet wins. So that gave you incentive to watch the Super Bowl and also watch the probably meaningless NBA game. All right, so basically let's try and figure out the chance each of those bets would win. Well, in order to do that we need to know something about a random variable called the Poisson random variable. And we're, what quantities follow a Poisson, basically, in a small length of time. Either one event, or zero event happens, you can't have more than one. Like for example, if you look at completed passes in the Super Bowl, in a time length of a few seconds, there will either be one completed pass or zero. And you look at an NBA game, in a small length of time there'll either be zero missed free throws by Shaq or one missed free throw by Shaq. And pretty much if that is true, the assumption that if in different times the events are independent and basically in a small length of time either zero or one events will happen, then you have a Poisson random variable and the Poisson dysfunction can give you the probabilities. And all you need is the mean. Like the normal random variable, you need the standard deviation. All you need to compute Poisson probabilities is the mean. And a lot of people will estimate the probability of a soccer team scoring zero, one, two, or three goals using the Poisson. We know how to predict the mean goals, a team scores in a soccer game from a previous video. I'm not sure Poisson works there. All you need to compute Poisson probabilities is the mean. So let's take a quick example. Suppose a teenager driver has an average of 0.3 accidents per year. What is the chance they'll have exactly one accident? And what is the chance of? Two or fewer accidents. Okay. Let's move this one down, okay. Exactly one accident would be Poisson. Okay you use .dist okay. Exactly one you put a one here, the mean is 0.3 and you put a false if you want the probability of exactly. So putting a one there with a false means the probability of exactly one accident. If you put a true it's the probability of less than or equal to one accident. So the chance of exactly one accident in a year would be 22%. Two or fewer accidents, you would do Poisson.dist. You'd put a 2, the mean is 0.3 and you'd put a true, that would be the chance of zero, one, or two accidents. And that's 99.6%. And you could check that a different way by finding the chance of zero, one or two accidents. And adding them up. So I do Poisson.dist. Here's the number of accidents, the mean is 0.3 and if I put a false I get the probability of exactly zero, one, or two. And if you add those up you should get 0.996. Okay, so armed with a Poisson random variable, we can handle the classic Trent Dilfer, Shaquille O'Neal props bet. Gosh I can't believe people bet on this. Okay so you look back on all the NFL games that year, I think Trent Dilfer averages 12 completions a game. You probably should adjust for the strength of the opponent there. He'd probably do less in the Super Bowl because they're probably playing a good defensive team. I didn't do that here. We'll talk about that more in the next video. And Shaquille O'Neal missed 7.3 free throws on the average per game. So let's assume the probabilities for Dilfer's number of completions and Shaq's missed free throws follow Poisson. And that makes sense because of the small length of time. It's going to be either zero completions or one or zero missed free throws or one. So I used the Poisson here to figure out the chance that Dilfer would have zero completions, one through, I don't know, 30, 40 completions. It gets very small. And I did the same for Shaquille O'Neal's missed free throws. I'll put in the mean, which is an F8 7.3, find the chance that Shaquille O'Neal will get zero through. Whatever, 40 missed free throws. Now how can Dilfer win the bet? Well, we add up all the ways this can happen. So for Dilfer to win the bet, we could have Dilfer, I'll just say Dilfer five with five completions. And then you'd need Shaq to have zero free throws missed because then the difference has to be greater than four. Or you could have Dilfer with six completions, Shaq would be less than or equal to one. You get the idea and these are what we call mutually exclusive. You can add them together. Dilfer with seven completions. You could have Shaq less or equal to two. Sorry, Dilfer with seven there. Okay, so you can add up all those probabilities. Okay, so like, what I did here was took the probably Dilfer got five completions, and Shaq zero, you just multiply them. E15 is the chance Dilfer gets five. F10 is the chance Shaq gets zero. You multiply those, and that's one way Dilfer can win with five completions. How about with six completions? Well you take the probability of six completions for Dilfer which is right here and then you multiply that times the chance of one or less which is adding up these two. Dilfer with seven completions, you take the chance of seven completions and you add up zero, one and two. Chance of less or equal to two. So you add all these up, you get a 51% chance Dilfer would win. Okay, the bet. The Dilfer bet would win. Now, what's the chance of a tie? Now I think I probably should have knocked down the Dilfer competitions mean a little bit, because the Super Bowl, you're going to be more conservative, you're probably playing a better defense so we'll see how that changes things in a minute. Okay, how can you have a tie? So this is Dilfer wins. So a tie could be all the ways you could have a tie. You could have Dilfer with four. And Shaq with zero. Or you could have Dilfer with five, Shaq with one. Dilfer with six, Shaq with two etc. So, Dilfer with four, you'd multiply by the chance that Shaq hit zero. So Dilfer four times Shaq zero. Dilfer with five, how could it be a tie? Dilfer five, Shaq one. Add those all up, you get 9%. And then take one minus the sum of these to get the chance Shaquille O'Neal wins. It looks like the Dilfer bet would be a really good bet. Now, if I had made this an 11, that may not be such a good bet. Oh, sorry, it's in the wrong place there. If I made that an 11 here. See then it becomes Shaquille O'Neal is a better bet and at probably about 11.5 it's going to be an even money bet. So, oh sorry, it's gotta go right here. And we, actually we could use Goal Seek here, I'm fiddling with this, which is something I told you not to do. But if I then put 11.4 here, we'll get it pretty close to even. That's pretty close to even. So Vegas probably had a good idea of what's going on here because in the Super Bowl, probably quarterbacks averaged a little bit fewer completions than they do during the regular season. And so that would make this bet really a very fair bet. If you think Trent Dilfer is going to get 11.4 completions per game, then it would be a very fair bet. So in the next video we'll talk about how to bet on how many touchdown passes Peyton Manning would throw in the Super Bowl against the Seahawks. How would you basically set probabilities for that. Again, that will use the Poisson and if you're going to set a bet on how many yards Peyton Manning might throw for in the Super Bowl, how would you set a fair number of yards for that? So we'll continue with this fascinating discussion in the next video.