Question

Out of $800$ families with $4$ children each, the expected percentage of families having almost $2$ girls, is:

• A. $68.75\%$
• B. $69.25\%$
• C. $62.50\%$
• D. None of these

Answer 1

The correct option is A $68.75\%$

Given: 800 families having 4 children each.

To find: the expected percentage of families having almost 2 girls

Sol:

Let X be the number of boys in a family, If p is the probability that a child is a boy and if genders are independent between children, the probability that a family ends up having k girls is

$P(X=x){=}^n{C}_k{p}^k(1-p{)}^{n-k}$

Here, $n=4,p=\frac{1}{2}$

Now we need to find for almost 2 girls, i.e.,

$P(X=0)+P(X=1)+P(X=2)⟹{}^4{C}_0{\left(\frac{1}{2}\right)}^0{\left(\frac{1}{2}\right)}^4{+}^4{C}_1{\left(\frac{1}{2}\right)}^1{\left(\frac{1}{2}\right)}^3{+}^4{C}_2{\left(\frac{1}{2}\right)}^2{\left(\frac{1}{2}\right)}^2⟹\frac{1}{16}+4\times \frac{1}{16}+6\times \frac{1}{16}=\frac{11}{16}=0.6875$

Hence the expected percentage of families having almost 2 girls, is $68.75\%$