Question

Alan decides to use a different strategy for the 1-in-6 wins game of Exercise 85. He keeps buying one 20-ounce bottle of the soda at a time until he gets a winner. (a) Find the probability that he buys exactly 5 bottles.

a. 0.0804
b. 0.1312
c. 0.1642
d. 0.1987

Answer 1

The probability that Alan buys exactly 5 bottles before winning is found using a geometric distribution formula, P(X=k) = q^(k-1) * p, which gives us 0.0804.

Step-by-step explanation:

To find the probability that Alan buys exactly 5 bottles before getting a winner in a 1-in-6 chance game, we employ a geometric distribution. The probability (p) of winning is 1/6, and the probability of losing (q) is 1 - 1/6 = 5/6. We want to find the probability that the first win occurs on the 5th trial, which means Alan must lose the first four times and then win on the fifth time.

The formula for the probability of the first success on the k-th trial in a geometric distribution is: P(X=k) = q^(k-1) * p. Plugging in the values, we get:

P(X=5) = (5/6)^4 * (1/6) = 0.0804

So, the probability that Alan buys exactly 5 bottles before getting a winner is 0.0804, which corresponds to option a.