A researcher wishes to conduct a study of the color preferences of new car buyers. Suppose that 50%
of this population prefers the color green. If 15
buyers are randomly selected, what is the probability that exactly a third of the buyers would prefer green? Round your answer to four decimal places.
To solve this problem, we can use the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
where:
P(X = k) is the probability of exactly k successes
n is the number of trials (buyers selected)
k is the number of successes (buyers who prefer green)
p is the probability of success in a single trial (probability of preferring green)
In this case, n = 15, k = 5 (one-third of 15), p = 0.50 (probability of preferring green).
Plugging in the values, we get:
P(X = 5) = (15 choose 5) * 0.50^5 * 0.50^10
P(X = 5) = (3003) * 0.03125 * 0.0009765625
P(X = 5) = 0.0947
Therefore, the probability that exactly a third of the buyers would prefer green is approximately 0.0947 when rounded to four decimal places.