Questions: A researcher wishes to conduct a study of the color preferences of new car buyers. Suppose that 50% of this population prefers the color red. If 12 buyers are randomly selected, what is the probability that exactly a half of the buyers would prefer red? Round your answer to four decimal places.

Solution Steps

A researcher is studying the color preferences of new car buyers, where \(50\%\) of the population prefers the color red. We want to find the probability that exactly half of the 12 randomly selected buyers prefer red.

We can model this scenario using a binomial distribution where:

• \(n = 12\) (the number of trials, or buyers),
• \(x = 6\) (the number of successes, or buyers preferring red),
• \(p = 0.5\) (the probability of success),
• \(q = 1 - p = 0.5\) (the probability of failure).

The probability of exactly \(x\) successes in \(n\) trials is given by the formula: \[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

Using the values defined: \[ P(X = 6) = \binom{12}{6} \cdot (0.5)^6 \cdot (0.5)^{12-6} \] Calculating this gives: \[ P(X = 6) = \binom{12}{6} \cdot (0.5)^{12} \] The binomial coefficient \(\binom{12}{6} = 924\), thus: \[ P(X = 6) = 924 \cdot (0.5)^{12} = 924 \cdot \frac{1}{4096} \approx 0.2256 \]

The probability that exactly 6 out of 12 buyers prefer red is: \[ P(X = 6) \approx 0.2256 \]

Final Answer