Final answer:
The distribution of the number of free throws the player takes until she misses can be described using a geometric distribution, making the assumption of independent events with a constant probability of making a free throw. The probability that the player will make 5 shots before she misses is 0.0181. The probability that she will make at most 5 shots before she misses can be found by summing the probabilities of making 1, 2, 3, 4, or 5 shots.
Step-by-step explanation:
(a) To describe the distribution of the number of free throws the player takes until she misses, we need to make the assumption that each free throw is an independent event and that the probability of making a free throw remains constant at 65% for each attempt. This assumption is necessary for modeling the situation using a geometric distribution.
(b) If the player makes 5 shots before she misses, we can use the geometric distribution formula to calculate the probability. The formula is P(X = k) = (1 - p)^(k-1) * p. So, P(X = 5) = (1 - 0.65)^(5-1) * 0.65 = 0.0181.
(c) To find the probability that she will make at most 5 shots before she misses, we need to sum the probabilities of making 1, 2, 3, 4, or 5 shots. P(X ≤ 5) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5). Using the geometric distribution formula, we can calculate each probability and sum them to get the final result.