Question

A probability teaser Suppose (as is roughly correct) that each child born is equally likely to be a boy or a girl and that the genders of successive children are independent. If we let BG mean that the older child is a boy and the younger child is a girl, then each of the combinations BB, BG, GB, and GG has a probability $0.25$Ashley and Brianna each have two children.

(a) You know that at least one of Ashley’s children is a boy. What is the conditional probability that she has two boys?

(b) You know that Brianna’s older child is a boy. What is the conditional probability that she has two boys?

Short answer

Part (a) $33.33\%$

Part (b) P (Two boy At least one boy) = $0.50$

Step-by-step solution

Part (a) Step 1. Given Information

At least one of Ashley's children is suffering from the disease.

Part (a) Step 2. Concept

Condition probability: $P\left(A\right|B)=\frac{P(AandB)}{P\left(B\right)}$

Part (a) Step 3. Calculation

Use the following formula to calculate the probability of a condition:

$P\left(A\right|B)=\frac{P(AandB)}{P\left(B\right)}$

$P(Twoboy||oldestisboy)=\frac{P(Twoboyandoldestisboy)}{P(oldestisboy)}=\frac{P(Twoboy)}{P(oldestisboy)}$

$P(Twoboys)=\frac{favourableoutcomes}{possibleoutcomes}=\frac{1}{4}$$=0.25$

$P(Atleastoneboy)=\frac{favourableoutcomes}{possibleoutcomes}=\frac{3}{4}=0.75$

As a result, the conditional probability is:

$P(Twoboy||Atleastoneboy)=\frac{P(Twoboys)}{P(Atleastoneboy)}=\frac{0.25}{0.75}=\frac{1}{3}\approx 0.3333=33.33\%$

As a result, the chances of her having two sons are $33.33\%$

Part (b) Step 1. Calculation

Condition probability: $P(A/B)=\frac{P(AandB)}{P\left(B\right)}$

$P(Twoboy||oldestisboy)=\frac{P(Twoboyandoldestisboy)}{P(oldestisboy)}=\frac{P(Twoboy)}{P(oldestisboy)}$

$P(Twoboys)=\frac{favourableoutcomes}{possibleoutcomes}=\frac{1}{4}=0.25$

$P(oldestisboy)=\frac{favourableoutcomes}{possibleoutcomes}=\frac{2}{4}=0.50$

As a result, the conditional probability is: $P(Twoboy||Atleastoneboy)=\frac{P(Twoboys)}{P(Atleastoneboy)}=\frac{0.25}{0.50}=0.50$

As a result, the conditional chance of her having two sons is $0.50$