Question

Given a family of two children (assume boys and girls equally likely, that is, probability 1/2 for each), what is the probability that both are boys? That at least one is a girl? Given that at least one is a girl, what is the probability that both are girls? Given that the first two are girls, what is the probability that an expected third child will be a boy?

Short answer

The required sample space is$\left(BB , BG , GB , GG\right)$ .

The probability that both are boys is $\frac{1}{4}$.

The probability that both are girls when at least one is a girl is $\frac{1}{3}$.

The probability of the third child being a boy is $\frac{1}{2}$.

Step-by-step solution

Significance of Probabilityof two chances

When there are only two possible outcomes, then the probability of each of them is $\frac{1}{2}$. For instance, when a coin is tossed then, there are $50\%$chances to get head or tail and hence the probability is $\frac{1}{2}$,

Creation of Sample 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, thus the possible outcomes for two children are 4 and it is expressed as,

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

Determination of the probability that both are boys

From the obtained sample space, it can be observed that only one case namely $\left(BB\right)$is the favourable outcome in which both are boy child, this implies that the probability that both are boys is $\frac{1}{4}$.

Thus, the probability that both are boys is $\frac{1}{4}$.

Determination of the probability that at least one is a girl

From the obtained sample space, it can be observed that only three cases namely $\left(GB , BG , GG\right)$are the favourable outcome in which there is at least one girl child, this implies that the probability that at least one is a girl is $\frac{3}{4}$.

Thus, the probability that at least one is a girl is $\frac{3}{4}$.

Determination of the probability that both are girls when at least one is a girl

From the obtained sample space, it can be observed that only three cases namely $\left(GB , BG , GG\right)$has the at least one girl, out of which only one case namely$\left(GG\right)$is the favourable outcome in which both are girl child ,this implies that the probability that both are girls when at least one is a girl is$\frac{1}{3}$

Thus, the probability that both are girls when at least one is a girl is$\frac{1}{3}$.

Determination ofthe probability that an expected third child will be a boy given that the first two children are girls

The probability of the third child being a boy is $\frac{1}{2}$as it does not depend whether the previous children were boy or girl. So, it does not depend on the obtained sample space.

Thus, the probability of the third child being a boy is $\frac{1}{2}$.