Expected Value in North Carolina’s Pick 4 Game In North Caroli...

Question

Expected Value in North Carolina’s Pick 4 Game In North Carolina’s Pick 4 lottery game, you can pay $1 to select a four-digit number from 0000 through 9999. If you select the same sequence of four digits that are drawn, you win and collect $5000.

d. Find the expected value.

e. If you bet $1 in North Carolina’s Pick 3 game, the expected value is Which bet is better in the sense of a producing a higher expected value: A $1 bet in the North Carolina Pick 4 game or a $1 bet in the North Carolina Pick 3 game?

Accepted Answer

Welcome back, everyone. In this problem, in a fictional lottery game called Lucky 5, players pay $2 to choose a five-digit number from 00000 to 99999. If their selected number exactly matches the five digit number drawn, they win $90,000. In another game, a $2 bet has an expected value of minus $1. Which option has a higher expected value, a $2 bet in the Lucky 5 game or the $2 bet in the other game? A says the $2 bet in the Lucky 5 game, B the $2 bet in the other game. C says both games have the same expected value, and the D says it cannot be determined. How can we figure out which option has a higher expected value? What do we know about expected values? Well, recall that the expected value, OK, is calculated by multiplying each outcome by its probability and adding these products. In other words, we'd multiply the probability of winning by the actual winnings, OK. And add it to the probability of losing multiplied by the actual amount lost. So in this case, we need to figure out what the probability of winning is, OK? And what, well, we already know how much we would win $90,000. And then we need to figure out the probability of losing and how much we'd lose in in whichever scenario. Now, first, let's focus on the winnings, OK? Overall, to solve this problem, OK, then we need to find how much possible outcomes can we have in our Lucky 5 game. Now in the lucky I have a game. OK, in the lucky 5 game. Then if we think about it, each digit in the 5 digit number can be anything from 0 to 9. So there are 10 possibilities for each digit. Thus, that means the total outcome, OK, or the total outcomes will be equal to 105, which is equal to 100,000. Therefore, this implies then, remember, they have to select exactly one number that matches the 5 digit number drawn. So the probability of winning, OK, is going to be equal to 1 divided by 100,000. Which thus means then that the probability of losing will be equal to 1 minus 1 divided by 100,000, which is equal to 99,999 divided by 100,000. OK, so a strong probability of losing here. Now when we win, OK, when we win or if we were to win, then like we said, we would need $90,000 but since you paid $2 to play, then your net gain would be $90,000 minus that $2 which is $89,998. On the other hand, for the loss, you don't collect any money, and since you paid $2 to play, then your net loss would be $2. Know that we have the probability of winning, the probability of losing, the winnings and the loss, then that means the expected value. Is going to be equal to 1 divided by 100,000 multiplied by 89,998 or winning scenario plus 99,999 divided by 100,000 multiplied by negative2 dollars or losing scenario. In this case, if we think about it here, this is going to be equal to 89,998 divided by 100,000 minus 199,998 divided by 100,000, which is gonna be equal to -10, 10,000 divided by $100,000 or negative dollar 10 cents. So for our lucky 5 game, the expected value is $-1.10. Remember, in the other game, the expected value was or $-1. So that means the $2 bet in the other game has a higher expected value of $-1 compared to the $2 bet in the Lucky 5 game of negative $1.10. Therefore, if we go back to our answer choices, that means that B. Is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Expected Value

Probability

Comparative Analysis

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