Questions: A researcher wishes to conduct a study of the color preferences of new car buyers. Suppose that 40% of this population prefers the color green. If 20 buyers are randomly selected, what is the probability that exactly a fifth of the buyers would prefer green? Round your answer to four decimal places.

Solution Steps

We are tasked with finding the probability that exactly \( x = 4 \) out of \( n = 20 \) new car buyers prefer the color green, given that the probability of a buyer preferring green is \( p = 0.40 \). The probability of not preferring green is \( q = 1 - p = 0.60 \).

The probability of exactly \( x \) successes in \( n \) trials for a binomial distribution is given by the formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

Substituting the values into the formula:

\[ P(X = 4) = \binom{20}{4} \cdot (0.40)^4 \cdot (0.60)^{20-4} \]

Using the binomial coefficient \( \binom{20}{4} \):

\[ \binom{20}{4} = \frac{20!}{4!(20-4)!} = \frac{20 \times 19 \times 18 \times 17}{4 \times 3 \times 2 \times 1} = 4845 \]

Now, calculate \( (0.40)^4 \) and \( (0.60)^{16} \):

\[ (0.40)^4 = 0.0256 \] \[ (0.60)^{16} \approx 0.010616 \]

Now, substituting these values back into the probability formula:

\[ P(X = 4) = 4845 \cdot 0.0256 \cdot 0.010616 \approx 0.035 \]

Final Answer