Question

A family has two children. If it is known that at least one child is a boy, then find the probability of both children being boys.

Hint:

Sample space {BB,BG,GB,GG}; restrict to outcomes with at least one boy, then find the fraction that are BB.

Answer 1

Step 1: Understanding the Concept:
With two children, each equally likely to be a boy (B) or girl (G), the sample space of birth orders is \(\{BB, BG, GB, GG\}\), each with probability \(1/4\). We need a conditional probability: given that "at least one is a boy" has already happened, what is the chance both are boys?

Step 2: Defining the events:
Let \(E\) = "both children are boys" \(=\{BB\}\). Let \(F\) = "at least one child is a boy" \(=\{BB,BG,GB\}\).

Step 3: Finding the probabilities:
\[ P(F) = \frac{3}{4},\qquad P(E\cap F) = P(\{BB\}) = \frac14 \]
(Note \(E\cap F = E\), since if both are boys, at least one is automatically a boy too.)

Step 4: Applying the conditional probability formula:
\[ P(E/F) = \frac{P(E\cap F)}{P(F)} = \frac{1/4}{3/4} = \frac13 \]

Final Answer:
The probability that both children are boys, given at least one is a boy, is \(\dfrac13\). \[ \boxed{\dfrac13} \]