At the Whistlestop Cafe, 15 of the 19 patrons are mechanics. If a sample of 8 is taken, what is the probability that exactly 5 patrons are mechanics? Round your answer to 4 decimal places.
To solve this problem, we need to use the concept of probability and the binomial probability formula.
The binomial probability formula states:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of obtaining exactly k successes
n is the total number of trials or sample size
k is the number of desired successes
p is the probability of success in a single trial
(1 - p) is the probability of failure in a single trial
In this case, we have a sample size of 8 and we want to find the probability of exactly 5 patrons being mechanics. We know that there are 15 mechanics out of a total of 19 patrons, so the probability of a single patron being a mechanic is p = 15/19, and the probability of a single patron not being a mechanic is (1 - p).
Using the binomial probability formula, we can calculate the probability as follows:
P(X = 5) = (8 C 5) * (15/19)^5 * (1 - 15/19)^(8 - 5)
Let's calculate this expression step by step:
(8 C 5) = 8! / (5! * (8-5)!) = (8 * 7 * 6) / (3 * 2 * 1) = 56
(15/19)^5 ≈ 0.4830 (rounded to 4 decimal places)
(1 - 15/19)^(8 - 5) ≈ 0.1039 (rounded to 4 decimal places)
P(X = 5) ≈ 56 * 0.4830 * 0.1039 ≈ 0.2608 (rounded to 4 decimal places)
Therefore, the probability that exactly 5 patrons are mechanics is approximately 0.2608.