Two children, one boy

Ms. Jackson has two children and at least one of them is a boy. What's the probability both children are boys?

What do you think?

Ms. Jackson has two children and at least one is a boy. P(both are boys)?

The sample space

Write out every possibility for two children, where order matters (older child first):

$\Omega = \{(B,B),\; (B,G),\; (G,B),\; (G,G)\}$

Each outcome has probability 1/4. The condition "at least one boy" eliminates $(G,G)$ , leaving three equally likely outcomes.

At least one boy

A mother with two children has at least one son. What's the probability both children are boys?

Sample space of two children:

GG (two girls) is eliminated. Of the 3 remaining, only BB has both boys → P = 1/3.

Conditioning on an event

When you condition on event $B$ , you restrict the sample space to outcomes where $B$ occurs and rescale probabilities so they sum to 1. If $A$ and $B$ are events, $P(A|B) = P(A \cap B) / P(B)$ .

Let $A$ = both children are boys, and $B$ = at least one child is a boy.

$P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{P(\{(B,B)\})}{P(\{(B,B),(B,G),(G,B)\})} = \frac{1/4}{3/4} = \frac{1}{3}$

A different question

Your colleague Ms. Parker also has two children. You see her walking with one of them, and that child is a boy. What's the probability both children are boys?

What do you think?

You see Ms. Parker with one child who is a boy. P(both children are boys)?

Why the answers differ

The two questions feel similar but they condition on different events.

The subtle difference

\text{Part A: at least one boy}

This is about the family as a whole. You learn that the pair isn't GG, but you don't know which child is the boy. Three outcomes survive: BB, BG, GB.

Step 1 of 4

You see a boy

You see a colleague walking with one of her two children, and that child is a boy. What's the probability both children are boys?

Sample space of two children:

You saw one specific child who is a boy. The other child is independent: P(boy) = 1/2.

The difference is between "at least one of the two satisfies a property" (which creates correlation) and "this particular one satisfies a property" (which leaves the other independent). How you acquire information determines the conditional probability.

Practice

A family has 3 children, each equally likely to be a boy or girl. Given at least one is a boy, what's P(all three are boys)? (decimal like 0.14)

1/3