Question

Given that the two numbers appearing on throwing two dice are different. Find the probability of the event 'the sum of numbers on the dice is 4'.

Answer 1

Consider the given events
A = Numbers appearing on two dice are different
B = The sum of the numbers on two dice is 4

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

\[\text{ Now } , \]

\[A \cap B = \left\{ \left( 1, 3 \right) \text{ and }  \left( 3, 1 \right) \right\}\]

\[ \therefore \text{ Required probability  } = P\left( B/A \right) = \frac{n\left( A \cap B \right)}{n\left( A \right)} = \frac{2}{30} = \frac{1}{15}\]