Question

If it is known that a woman has two children and she has at least one girl child, then the probability that both children are girls is

• A. \(\frac13\)
• B. \(\frac14\)
• C. \(\frac12\)
• D. \(\frac23\)

Hint:

Whenever a condition such as ``at least one'' is given, first reduce the sample space accordingly and then calculate probability within the reduced sample space.

Answer 1

The Correct Option is A

Step 1: List all possibilities.
A family with two children, in birth order, can be $BB$, $BG$, $GB$, or $GG$. Each is equally likely.

Step 2: Apply the given condition.
We are told there is at least one girl. So we throw away the outcome $BB$, which has no girl.

Step 3: Write the reduced list.
After removing $BB$, the remaining equally likely outcomes are $BG$, $GB$, $GG$. That is three outcomes.

Step 4: Find the favourable outcome.
We want both children to be girls, which is the single case $GG$.

Step 5: Form the conditional probability.
Probability equals favourable over total within the reduced list: \[ P=\frac{1}{3}. \]

Step 6: Why it is not one half.
It feels like it should be $\dfrac12$, but the condition keeps three cases, not two, because $BG$ and $GB$ are different. So the chance is smaller.

Step 7: State the answer.
The required probability is $\dfrac13$. \[ \boxed{\dfrac13} \]