# 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?

(a) $\frac{3}{8}$

(b) $\frac{7}{9}$

(c) $\frac{2}{3}$

(d) $\frac{5}{6}$

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Answer: (C)  $\frac{2}{3}$

Let S be the sample space of rolling a dice.

Then, S = {1, 2, 3, 4, 5, 6} ⇒ n(S) = 6

Let A: Event of rolling an odd number and

B: Event of rolling a prime number.

Then, A = {1, 3, 5} ⇒ n(A) = 3

B = {2, 3, 5} ⇒ n(B) = 3

A ∩ B = {3, 5} ⇒ n(A ∩ B) = 2

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

Now, P(Rolling a prime number, if the outcome is an odd number)

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