Question

Assuming that for a husband wife couple the chances of their child being a boy or a girl are the same, the probability of their two children being a boy and a girl is

• A. $\frac{1}{4}$
• B. $1$
• C. $\frac{1}{2}$
• D. $\frac{1}{8}$

Answer 1

The correct option is C

$\frac{1}{2}$

Explanation for the correct answer:

Step 1: According to the question.

Let p be the probability of being boy or girl, $p=\frac{1}{2}$

Let q be the probability of failure,

$$\begin{gathered} q=1-p \\ =1-\frac{1}{2} \\ =\frac{1}{2} \end{gathered}$$

Let n be the number of children, $n=2$

Let x be the number of boys, $x=1$

Step 2: Applying Binomial distribution formula.

The formula of binomial distribution is

$P\left(x\right)=n{C}_x{p}^x{q}^{n-x}$

On substituting the above values,

$$\begin{gathered} P(boyandgirl)=2C_1{\left(\frac{1}{2}\right)}^1{\left(\frac{1}{2}\right)}^{2-1} \\ =2\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)...\mathbf{}\left[\because 2C_1=2\right] \\ =\frac{1}{2} \end{gathered}$$

Thus using binomial distribution, the probability of their two children being a boy and a girl is $\frac{1}{2}$

Hence, option (C) $\frac{\mathbf{1}}{\mathbf{2}}$ is the correct answer.