Suppose that a family has 5 children and that the probability of having a girl is $\frac { 1 } { 2 }$ . What is the probability of having no more than three boys?
A) 0.8125
B) 0.9688
C) 0.5000
D) 0.3125

Question

Quizplus · Accepted Answer

The probability of having no more than three boys includes the scenarios of having 0, 1, 2, or 3 boys. Since the probability of having a boy or a girl is $\frac{1}{2}$, we can calculate the probability for each scenario and sum them up. The probability of having exactly k boys out of 5 children is given by the binomial probability formula $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$, where $n=5$, $p=\frac{1}{2}$, and $k$ is the number of boys. Calculating for $k=0, 1, 2, 3$ and summing gives $P(X\leq3) = \binom{5}{0}\left(\frac{1}{2}\right)^5 + \binom{5}{1}\left(\frac{1}{2}\right)^5 + \binom{5}{2}\left(\frac{1}{2}\right)^5 + \binom{5}{3}\left(\frac{1}{2}\right)^5 = 0.8125$.

Suppose That a Family Has 5 Children and That the Probability