Expected payoff example: lottery ticket

- [Instructor] We're told a Pick 4 lottery game involves drawing four numbered balls from separate bins, each containing balls labeled from 0 to 9. So there are 10,000 possible selections in total. For example, you could get a 0, a 0, a 0 and a 0, a 0, a 0, a 0 and a 1, all the way up to 9,999, four nines. Players can choose to play a straight bet, where the player wins if they match all four digits in the correct order. The lottery pays $4,500 on a successful $1 straight bet. Let X represent a player's net gain on a $1 straight bet. Calculate the expected net gain. And they say, hint, the expected net gain can be negative. So why don't you pause this video and see if you can calculate the expected net gain? All right. So there's a couple of ways that we can approach this. One way is to just think about the two different outcomes. There's a scenario where you win with your straight bet. There's a scenario where you lose with your straight bet. Now let's think about the net gain in either one of those scenarios. The scenario where you win, you pay $1, we know it's a $1 straight bet, and you get $4,500. So what's the net gain? So it's going to be $4,500 minus one. So your net gain is going to be $4,499. Now what about the net gain in the situation that you lose? Well, in the situation that you lose, you just lose a dollar. So this is just going to be negative $1 right over here. Now let's think about the probabilities of each of these situations. So the probability, so the probability of a win we know is 1 in 10,000, 1 in 10,000. And what's the probability of a loss? Well, that's going to be 9,999 out of 10,000. And then our expected net gain is just going to be the weighted average of these two. So I could write our expected net gain is going to be 4,499 times the probability of that, 1 in 10,000 plus negative 1 times this, so that I could just write that as minus 9,999 over 10,000. And so this is going to be equal to, let's see, it's going to be 4,499 minus 9,999, all of that over 10,000. And let's see, this is going to be equal to negative 5,500 over 10,000, negative 5,500 over 10,000, which is the same thing as negative 55 over 100, or I could write it this way. This is equal to negative .55. I could write it this way, 0.55. So that's one way to calculate the expected net gain. Another way to approach it is to say, all right, what if we were to get 10,000 tickets? What is our expected net gain on the 10,000 tickets? Well, we would pay $10,000 and we would expect to win once. It's not a guarantee, but we would expect to win once. So expect 4,500 in payout. And so you would then, let's see, you would have a net gain of, it would be negative $5,500, negative $5,500. Now this is the net gain when you do 10,000 tickets. Now, if you wanted to find the expected net gain per ticket you would then just divide by 10,000. And if you did that, you would get exactly what we just calculated the other way. So any way you try to approach this this is not a great bet.