# A die is rolled. If the outcome is an odd number, what is the probability that it is a prime number ?

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A die is rolled. If the outcome is an odd number, what is the probability that it is a prime number ?

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S = {1, 2, 3, 4, 5, 6} ⇒ n(S) = 6

Let A : Event of getting an odd number

B : Event of getting a prime number

A = {1, 3, 5} ⇒ n(A) = 3

B = {2, 3, 5} ⇒ n(B) = 3

A ∩ B = {3, 5} ⇒ n(A ∩ B) = 2

∴ P(A) = $\frac{n(A)}{n(S)} =\frac{3}{6} =\frac{1}{2}$

P(B) = $\frac{n(B)}{n(S)} =\frac{3}{6} =\frac{1}{2}$

P(A ∩ B) =$\frac{n(A\,\cap\,B)}{n(S)}=\frac{2}{6}=\frac{1}{3}$

∴ P(B/A) = $\frac {P(A\,\cap\,B)}{P(A)}=\frac{\frac{1}{3}}{\frac{1}{2}}= \frac{2}{3}$ .