Question

A pair of dice is thrown. Find the probability of getting 7 as the sum if it is known that the second die always exhibits a prime number.

Answer 1

Consider the given events
A = A prime number appears on second die.
B = The sum of the numbers on two dice is 7.

Clearly,
A = {(1, 2), (1, 3), (1, 5), (2, 2), (2, 3)(2, 5), (3, 2), (3, 3), (3, 5) (4, 2), (4, 3), (4, 5),(5, 2), (5, 3), (5, 5), (6, 2), (6, 3),(6, 5)}
B = {(2, 5), (5, 2), (3, 4), (4, 3), (1, 6), (6, 1)}

$$\begin{gathered} \mathrm{Now}, \\ A\cap B=\{(2,\; 5),\; (5,\; 2),(4,\; 3)\} \end{gathered}$$

$\therefore \mathrm{Required}\mathrm{probability}=P\left(B/A\right)=\frac{n\left(A\cap B\right)}{n\left(A\right)}=\frac{3}{18}=\frac{1}{6}$