A researcher wishes to conduct a study of the color preferences of new car buyers. Suppose that 30%

Question

of this population prefers the color brown. If 12
buyers are randomly selected, what is the probability that exactly 5
buyers would prefer brown? Round your answer to four decimal places.

Answer 1

To calculate the probability that exactly 5 out of 12 buyers would prefer brown, we can use the binomial probability formula:

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

Where:
n = total number of trials (12 buyers)
k = number of successful outcomes (5 buyers preferring brown)
p = probability of success on any given trial (30% or 0.3)

Plugging in the values:
P(X = 5) = (12 choose 5) * 0.3^5 * (1-0.3)^(12-5)

Calculating (12 choose 5):
(12 choose 5) = 792

Now calculate the probability:
P(X = 5) = 792 * 0.3^5 * 0.7^7
P(X = 5) = 792 * 0.00243 * 0.08235
P(X = 5) = 1.8394

So, the probability that exactly 5 out of 12 buyers would prefer brown is approximately 0.1839 or 18.39%.