Probability Question 9

The probability of winning a certain game is 0.5. If at least 70 percent of the games in a series of $n$ games are won, the player wins a prize. If the possible choices for $n$ are

n=10, n=20, \text{ and } n=100,

which value of $n$ should the player choose in order to maximize the probability of winning a prize?

n=10

only

n=100

only

n=10

n=100

only; the probabilities are the same.

n=10

n=20

n=100

; the probabilities are the same.

Approach

The central limit theory implies that the greater then number of games, the more likely the sample average would approach the true average. In our situation, as $n$ gets sufficienly large, the percent of times we win will approach 50. To have the highest chance of winning, we should choose the lowest number of games. This way, due to variation and a bit of luck, we can win 7 out of the 10 games.