Question

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

Answer 1

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)

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

\(=\frac{2}{3}\)