A couple has two children, (i) Find the probability that both children are males, if it is known that at least one of the children is male. (ii) Find the probability that both children are females, if it is known that the elder child is a female.

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A couple has two children, (i) Find the probability that both children are males, if it is known that at least one of the children is male.
(ii) Find the probability that both children are females, if it is known that the elder child is a female.

CBSE | Class 12 | Maths | NCERT | Probability

A couple has two children, (i) Find the probability that both children are males, if it is known that at least one of the children is male.
(ii) Find the probability that both children are females, if it is known that the elder child is a female.

Solution:
(i) According to the question, if the couple has two children then the sample space is:

$S={(b, b),(b, g),(g, b),(g, g)}$

Assume A denotes the event of both children having male children, and B denotes the event of at least one of the male children having male children.

As a result, we have:

$A \cap B={(b, b)}$

$P(A \cap B)=\frac{1}{4}$

$P(A)=1 / 4$

$P(B)=3 / 4$

Hence, $P(A \mid B)=\frac{P(A \cap B)}{P(B)}$

By substituting the values we get

$=$

$=1 / 3$

(ii) Assume that $C$ represents the event of having both female children and D represents the event of having a female elder child.
$\therefore \mathrm{C}={(\mathrm{g}, \mathrm{g})}$