A researcher wishes to conduct a study of the color preferences of new car buyers. Suppose that 40%
of this population prefers the color black. If 18
buyers are randomly selected, what is the probability that exactly 8
buyers would prefer black? Round your answer to four decimal places.
To calculate the probability that exactly 8 buyers would prefer black, we can use the probability mass function for a binomial distribution:
P(X = 8) = (n choose k) * p^k * (1-p)^(n-k)
where:
n = total number of buyers selected = 18
k = number of buyers preferring black = 8
p = probability of preference for black = 0.40
Plugging in the values, we get:
P(X = 8) = (18 choose 8) * 0.40^8 * 0.60^10
Calculating the binomial coefficient:
(18 choose 8) = 18! / (8! * (18-8)!) = 43758
Now, plug in everything and calculate the probability:
P(X = 8) = 43758 * 0.40^8 * 0.60^10
P(X = 8) ≈ 0.1644
Therefore, the probability that exactly 8 buyers would prefer black is 0.1644 (rounded to four decimal places).