UPSC NDA I 2024 Mathematics Paper

There are two ways the man can select two balls of the same color:Scenario 1: Both balls are blackThe probability of selecting one black ball is $\frac{5}{9}$. After taking out one black ball, there are only 4 black balls left and 8 total balls. So, the probability of selecting another black ball is $\frac{4}{8}$.The probability of scenario 1 is then: $\frac{5}{9} \times \frac{4}{8} = \frac{5}{18}$.Scenario 2: Both balls are whiteThe probability of selecting one white ball is $\frac{4}{9}$. After taking out one white ball, there are only 3 white balls left and 8 total balls. So, the probability of selecting another white ball is $\frac{3}{8}$.The probability of scenario 2 is then: $\frac{4}{9} \times \frac{3}{8} = \frac{1}{6}$.Total ProbabilityTo get the probability of selecting two balls of the same color, we add the probabilities of the two scenarios:$\frac{5}{18} + \frac{1}{6} = \frac{8}{18} = \frac{4}{9}$