Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

You know there are 3 boys and an unknown number of girls in a nursery at a hospital. Then a woman gives birth a baby, but you do not know its gender, and it is placed in the nursery. Then a nurse comes in a picks up a baby and it is a boy. Given that the nurse picks up a boy, what is the probability that the woman gave birth to a boy?

So I've narrowed it down (taking the number of girls to be $n$) to:

$$\begin{align*} &P(\text{woman had a boy} \mid \text{nurse picked up boy)}\\ &\qquad= \frac{P(\text{nurse picked up boy} \mid \text{woman had a boy}) P(\text{woman had boy})}{P(\text{nurse picked up boy})}\\ &\qquad=\frac{\frac{4}{(4+n)}\cdot0.5}{3/(3+n)}\;, \end{align*}$$

but there's no way for me to get an analytical answer that makes sense. The correct answer is $\frac{4}{7}$.

share|cite|improve this question

Your denominator would be $P(\text{nurse picked up a boy})$ before the new baby was ever born. You need the probability that the nurse picked up a boy marginalized across both genders that the new baby could be, which is to say $$.5\frac{3}{4+n}+.5\frac{4}{4+n}$$ This should be the denominator since there are definitely $4+n$ children in the nursery now, and either $3$ or $4$ of them are boys.

Now to get $4/7$ is just to cancel the $.5/4+n$ from numerator and denominator.

share|cite|improve this answer

Let $E_1$ and $E_2$ be two events defined as

$E_1:$ boy is born

$E_2:$ girl is born

Let say that initially there were $n$ girls.

$B:$ boy picked up by nurse

$G:$ girl picked up by nurse


share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.