Expected Value Intro

Casino Games Activity 9

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Probability What is expected value? If you figure out how much money you’d win or lose on average, per try, for a particular game over the long run, you’ve calculated the expected value. Expected value is calculated by multiplying each possible outcome (gain a dollar, lose a dollar) by its probability, then adding all of those products. Example 1: Let’s say you are playing a game where you get paid $1 per flip if a fair coin comes up heads, and you lose $1 if the coin comes up tails. Calculate the expected value. There are two possible outcomes, +$1 and -$1. Because the coin is fair, the probability of flipping heads is 0.5 just like the probability of flipping tails is 0.5. To find the expected value, multiply each outcome by its probability and then sum all those products: (+$1)(0.5) + ("$1)(0.5) = $0.50 + ("$0.50) = $0 . In the long run, you’d break even, because the expected value is zero. When the expected value is zero, it is called a fair game.

! Example 2: Your friend has a coin hidden under 1 of 3 cups. Your friend switches the cups around, then asks you to guess which cup the coin is under. Your friend says if you pick correctly, he’ll give you $12. If you pick incorrectly, you only have to pay $9. If your friend is willing to play this game over and over again, is this a good moneymaking enterprise for you? Assume that the cups move too fast for you to follow, and you just have to guess. Also assume your friend isn’t cheating like con men do when they play this game. Calculate the expected value to find out. There are two possible outcomes, +$10 and -$9. Because you’re guessing, the probability of winning is 1/3 which the probability of losing is 2/3.

# 1& # 2& ( + ("$9)% ( = $4 + ("$6) = "$2 $ 3' $ 3'

(+$12)%

Because the expected value is negative, you’d lose money in the long run by playing this game.

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