Question

A student claims in Problem 1.5 that if one child is a girl, the probability thatboth are girls is $\frac{\mathbf{1}}{\mathbf{2}}$. Use appropriate sample spaces to show what is wrong withthe following argument: It doesn’t matter whether the girl is the older child or theyounger; in either case the probability is $\frac{\mathbf{1}}{\mathbf{2}}$that the other child is a girl.

Short answer

The probability of the second child being a girl is$\frac{1}{3}$ when another child is a girl.

Step-by-step solution

Definition of Sample Space

Sample space of any experiment is the set of all possible mutually exclusive events.Sample space can be uniform or non-uniform, it depends on the probability of each event in the sample space.

Creation ofsample space

A child being a girl or a boy does not depend on whether the previous child was a girl or a boy.

Let G represents girl child and B represents Boy child, so, the possible outcomes for two children are 4 that are as follows,

$\left(BB , BG , GB , GG\right)$

Analysing the argument

When one child is a girl, this makes out of the events and the sample space becomes $\left(BG , GB , GG\right)$and each has a probability of $\frac{1}{3}$, this implies that the probability of other child being a girls is$\frac{1}{3}$ when the other child is also a girl.

Thus, the given argument is incorrect.