Questions: Assume that male and female births are equally likely and that the birth of any child does not affect the probability of the gender of any other children. Find the probability of exactly three boys in ten births. Round the answer to the nearest thousandth. A. 0.030 B. 0.117 C. 15.000 D. 0.300

Solution Steps

We need to find the probability of having exactly 3 boys in 10 births, assuming that the probability of a boy or a girl is equal, i.e., \( p = 0.5 \) and \( q = 1 - p = 0.5 \).

The probability of getting exactly \( x \) successes (boys) in \( n \) trials (births) can be calculated using the binomial probability formula:

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

Where:

• \( n = 10 \) (total births)
• \( x = 3 \) (number of boys)
• \( p = 0.5 \) (probability of a boy)
• \( q = 0.5 \) (probability of a girl)

The binomial coefficient \( \binom{n}{x} \) is calculated as follows:

\[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \]

Now we can substitute the values into the formula:

\[ P(X = 3) = \binom{10}{3} \cdot (0.5)^3 \cdot (0.5)^{10-3} \]

Calculating this gives:

\[ P(X = 3) = 120 \cdot (0.5)^3 \cdot (0.5)^7 = 120 \cdot (0.5)^{10} = 120 \cdot \frac{1}{1024} = \frac{120}{1024} = 0.1171875 \]

Rounding to three decimal places, we find:

\[ P(X = 3) \approx 0.117 \]

Final Answer