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 probability question, you'll need to use the concept of hypergeometric distribution. The hypergeometric distribution is used when you have a sample drawn from a finite population without replacement. In this case, you have a sample of 8 patrons drawn from a total of 19.
The probability of exactly 5 patrons being mechanics can be calculated using the formula:
P(X = k) = (C(m, k) * C(N - m, n - k)) / C(N, n)
where:
P(X = k) is the probability of exactly k successes (in this case, 5 mechanics),
C(a, b) is the combination function of a choose b,
m is the number of mechanics (15),
N is the total number of patrons (19),
n is the sample size (8), and
k is the number of mechanics in the sample (5).
Now, let's substitute the values into the formula:
P(X = 5) = (C(15, 5) * C(19 - 15, 8 - 5)) / C(19, 8)
C(15, 5) = 3003 (15 choose 5)
C(4, 3) = 4 (4 choose 3)
C(19, 8) = 75582 (19 choose 8)
P(X = 5) = (3003 * 4) / 75582
P(X = 5) ≈ 0.1585
Therefore, the probability that exactly 5 out of the 8 patrons are mechanics is approximately 0.1585 when rounded to four decimal places.