Question

A dice is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once?

Answer 1

Consider the given events.
A = 4 appears on the die
B = The sum of the numbers on two dice is 6.

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

\[\text{ Now } , \]
\[A \cap B = \left\{ \left( 2, 4 \right) \text{ and }  \left( 4, 2 \right) \right\}\]
\[ \therefore \text{ Required probability } = P\left( A/B \right) = \frac{n\left( A \cap B \right)}{n\left( B \right)} = \frac{2}{5}\]