To solve this problem, we need to calculate the probability that exactly one out of three randomly selected people is a woman. Assuming the probability of selecting a woman is given by the decimal 0.416667, we can use the binomial probability formula. The formula for the probability of getting exactly \( k \) successes (in this case, selecting a woman) in \( n \) trials (people selected) is given by:
\[ P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k} \]
where \( \binom{n}{k} \) is the binomial coefficient, \( p \) is the probability of success on a single trial, and \( n \) is the number of trials.
We are given the probability of selecting a woman, \( p = 0.4167 \). We need to find the probability that exactly one out of three randomly selected people is a woman. The number of trials is \( n = 3 \), and the number of successes (selecting a woman) is \( k = 1 \).
The probability of getting exactly \( k \) successes in \( n \) trials is given by the binomial probability formula:
\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}
\]
Substituting the values, we have:
\[
P(X = 1) = \binom{3}{1} \cdot (0.4167)^1 \cdot (1 - 0.4167)^{3-1}
\]
The binomial coefficient \(\binom{3}{1}\) is calculated as:
\[
\binom{3}{1} = \frac{3!}{1!(3-1)!} = 3
\]
Substitute the values into the formula:
\[
P(X = 1) = 3 \cdot 0.4167 \cdot (1 - 0.4167)^2
\]
Calculate the probability:
\[
P(X = 1) = 3 \cdot 0.4167 \cdot 0.5833^2
\]
\[
P(X = 1) \approx 0.4253
\]
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