Expected payoff example: lottery ticket | Probability & combinatorics | Khan Academy

We're told a pick four lottery game involves drawing four numbered balls from separate bins, each containing balls labeled from zero to nine. So, there are ten thousand possible selections in total. For example, you could get a zero, a zero, a zero, and a zero; a zero, a zero, a zero, and a one; all the way up to nine thousand nine hundred and ninety-nine, 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 five hundred dollars on a successful one dollar straight bet. Let x represent a player's net gain on a one dollar 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, and 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 one dollar, we know it's a one dollar straight bet—and you get four thousand five hundred dollars. So what's the net gain? So it's going to be four thousand five hundred dollars minus one. So your net gain is going to be four thousand four hundred and ninety-nine dollars.

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 going to be negative one dollar right over here. Now, let's think about the probabilities of each of these situations.

So the probability of a win we know is one in ten thousand, one in ten thousand. And what's the probability of a loss? Well, that's going to be nine thousand nine hundred and ninety-nine out of ten thousand. And so 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 four thousand four hundred and ninety-nine times the probability of that one in ten thousand, plus negative one times this. So that I could just write that as minus nine thousand nine hundred and ninety-nine over ten thousand.

And so this is going to be equal to, let's see, it's going to be four thousand four hundred ninety-nine minus nine thousand nine hundred and ninety-nine, all of that over ten thousand. And let's see, this is going to be equal to negative five thousand five hundred over ten thousand, negative five thousand five hundred over ten thousand, which is the same thing as negative fifty-five over one hundred or I could write it this way: this is equal to negative fifty-five hundredths. 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 ten thousand tickets? What is our expected net gain on the ten thousand tickets? Well, we would pay ten thousand dollars, and we would expect to win once. It's not a guarantee, but we would expect to win once, so expect four thousand five hundred in payout.

And so you would then, let's see, you would have a net gain of it would be negative five thousand five hundred dollars, negative five thousand five hundred dollars. Now this is the net gain when you do ten thousand tickets. Now, if you wanted to find the expected net gain per ticket, you would then just divide by ten thousand.

And if you did that, you would get exactly what we just calculated the other way. So anyway, you try to approach this, this is not a great bet.