There are 7 children. You are told that the youngest child is a boy. The…

Concept: When events are independent — like the gender of each child, each with probability 1/2 of being a boy or a girl — the probability that a specific set of n independent events all occur is the product of their individual probabilities, i.e. (1/2)n. If one outcome is already given or fixed, it drops out of the calculation and only the remaining independent events matter.

Application:

1. There are 7 children in total, and we are told the youngest child is already a boy — this outcome is fixed and given, not random anymore.

2. So only the remaining 6 children's genders are still independent unknowns.

3. For all 7 children to be boys, each of these remaining 6 children must independently be a boy.

4. The probability that one child is a boy is 1/2, so for 6 independent children: 1/2 × 1/2 × 1/2 × 1/2 × 1/2 × 1/2 = (1/2)6 = 1/64.

5. So the probability that all of them are boys, given the youngest is a boy, is 1 in 64.

Cross-check: Using the conditional probability formula, P(all 7 boys ∣ youngest is boy) = P(all 7 boys) / P(youngest is boy) = (1/2)7 / (1/2) = (1/2)6 = 1/64. This confirms the same result by an independent method.