Probability that the first three of the five children in the family are boys?

Question

Problem: In a family of 5 children, what is the probability that the first three are boys?

Attempt: I used binomial probability. Where the n = 5, r = 3, success = 0.5 and failure = 0.5.

$P = (5C3)(0.5)^3(0.5)^2 $ = $\frac{5}{16}$

But the answer is 1/8 which I don't how it was solved.

Question: How to answer this kind of problems? I set 0.5 as there are two probabilities of a child's gender (male or female). Any help would be appreciated.

Answer 1

In your answer you did not calculate the probability that the first $3$ children are boys, but the probability that there are exactly $3$ boys among the $5$ children.

The probability that the first $3$ children are boys is:$$\frac12\frac12\frac12$$by independence.

Here the first factor stands for the probability that the first child is a boy, the second factor stands for the probability that the second child is a boy and third factor stands for the probability that the third child is a boy.

Answer 2

The probability for a boy is $\frac{1}{2}$. Hence, the probability for giving birth to three boys in a row is $\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}=\frac{1}{8}$. You calculated that the probability for exactly three boys in 5 children, not that the first three are boys.