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: List the sample space explicitly and simply count outcomes, instead of naming events with formal set notation first:
The four equally likely birth orders for two children are BB, BG, GB, GG.

Step 2: Cross out every outcome that does NOT have at least one boy:
GG has no boy at all, so it is ruled out by the given information. The remaining possible outcomes are BB, BG, GB — three outcomes, all equally likely among themselves now.

Step 3: Count how many of these three remaining outcomes have both children as boys:
Only BB qualifies as "both boys," which is 1 out of the 3 remaining equally-likely outcomes.

Step 4: Convert the count to a probability:
$P(\text{both boys} \mid \text{at least one boy}) = \dfrac{1}{3}$.

Final Answer:
The required probability is $\dfrac13$. \[ \boxed{\dfrac13} \]