Question
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.
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.
Answers
GPT 3.5
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.
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.