A pair of dice is thrown and the sum of the numbers appearing on the dice is observed to be 7. Find the probability that the number 5 has appeared on atleast one die.

$\frac{1}{3}$

The correct answer is Option (2) → $\frac{1}{3}$ ##

Let $E = \text{event that 5 has appeared on atleast one die}$

$∴E = \{(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 5), (4, 5), (3, 5), (2, 5), (1, 5)\}$

Let $F = \text{event that sum of no. on die is 7.}$

$∴F = \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\}$

$E \cap F = \{(2, 5), (5, 2)\}$

$∴n(E \cap F) = 2 \text{ and } n(F) = 6$

$\text{Now, } P\left(\frac{E}{F}\right) = \frac{P(E \cap F)}{P(F)} = \frac{n(E \cap F)}{n(F)} = \frac{2}{6} = \frac{1}{3}$