What is the probability of a family with 6 children having 3 boys and 3 girls?
A) $\frac { 5 } { 16 }$
B) $\frac { 11 } { 16 }$
C) $\frac { 1 } { 2 }$
D) 0
E) $\frac { 3 } { 64 }$

Question

Quizplus · Accepted Answer

The probability of having 3 boys and 3 girls in a family of 6 children can be calculated using the binomial coefficient, which gives us the number of ways to choose 3 boys out of 6 children, multiplied by the probability of each of those outcomes. The formula for the binomial coefficient is $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n$ is the total number of children, and $k$ is the number of boys (or girls, since they are equal in this case). Thus, we have $\binom{6}{3} = \frac{6!}{3!3!} = 20$ ways to have 3 boys and 3 girls. The probability of each child being a boy or a girl is $\frac{1}{2}$, assuming equal probability for each gender. Therefore, the probability of having 3 boys and 3 girls is $20 \times (\frac{1}{2})^6 = \frac{20}{64} = \frac{5}{16}$.

What Is the Probability of a Family with 6 Children 516\frac { 5 } { 16 }165​

What Is the Probability of a Family with 6 Children $\frac { 5 } { 16 }$