UPSC NDA 2 2022 Mathematics Solved Paper

Concept:The problem involves Binomial distribution. In $n$ independent trials, if the probability of success is $p$ and failure is $q = 1-p$, then the probability of exactly $r$ successes is ${}^{n}C_{r} p^{r} q^{n-r}$.Explanation:Given: On average, 1 out of 5 ships is sunk. So, probability of sinking (failure) is $q = \frac{1}{5}$. Thus, probability of arriving safely (success) is $p = 1 - q = 1 - \frac{1}{5} = \frac{4}{5}$. We need the probability that exactly 3 out of 5 ships arrive safely. Here, $n = 5$, $r = 3$, $p = \frac{4}{5}$, $q = \frac{1}{5}$. Using the binomial formula: Probability = ${}^{5}C_{3} \left(\frac{4}{5}\right)^3 \left(\frac{1}{5}\right)^{5-3}$. ${}^{5}C_{3} = \frac{5!}{3!2!} = 10$. $\left(\frac{4}{5}\right)^3 = \frac{64}{125}$, and $\left(\frac{1}{5}\right)^2 = \frac{1}{25}$. Thus, probability = $10 \times \frac{64}{125} \times \frac{1}{25} = \frac{640}{3125} = \frac{128}{625}$.Answer:$\frac{128}{625}$ (Option D).