If sharma family has two children. What is the probability that both children are boys given that at least one of them is boy?

Concept:

Let A and B be any two events associated with a random experiment. Then, the probability of occurrence of an event A under the condition that B has already occurred such that P(B) ≠ 0, is called the conditional probability and denoted by P(A | B)

i.e \(P\;\left( {A|\;B} \right) = \frac{{P\;\left( {A\; ∩ B} \right)}}{{P\left( B \right)}}\)

Similarly, \(P\left( {B\;|\;A} \right) = \frac{{P\left( {A\; ∩ B} \right)}}{{P\left( A \right)}},\;where\;P\left( A \right) \ne 0\)

Calculation:

Let b stand for boy and g for girl.

The sample space S = {(b, b), (b, g), (g, g), (g, b)}

Let A = Both the children are boys

Let B = At least one of the child is boy.

i.e A = {(b, b)}, B = {(b, b), (b, g), (g, b)} and A ∩ B = {(b, b)}

⇒ P (B) = 3/4 and P (A ∩ B) = 1/4

As we know that, \(P\;\left( {A|\;B} \right) = \frac{{P\;\left( {A\; ∩ B} \right)}}{{P\left( B \right)}}\)

⇒ P (A | B) = 1/3