Answer to Question #174867 in Statistics and Probability for nivil

Question

1.     The students at a swimming class consists of 10 girls, 30 boys and 10 adults. After a month lesson, 3 of the girls, 10 of the boys and 3 of the adults were able to swim. If a student is chosen at random from this class and is found to be able to swim, what is the probability that the student is a boy?

Answer 1

Let $A = \{Student \ is \ able \ to \ swim\}$

$H_i = \{Chosen \ at \ random \ student \ is \ i\} \ \forall i \in \{boy, girl, adult\}$

Then using Bayes formula:

$Pr(H_{boy} | A) = \frac{Pr(A|H_{boy})Pr(H_{boy})}{Pr(A)} = \frac{Pr(A|H_{boy})Pr(H_{boy})}{\sum_{i}Pr(A|H_i)Pr(H_i)}$

Let's find these probabilities:

$Pr(A|H_{boy}) = \frac{10}{30} = \frac13$

$Pr(A|H_{girl}) = \frac{3}{10} = 0.3$

$Pr(A|H_{adult}) = \frac{3}{10} = 0.3$

$Pr(H_{boy}) = \frac{30}{50} = 0.6$

$Pr(H_{girl}) = \frac{10}{50} = 0.2$

$Pr(H_{adults}) = \frac{10}{50} = 0.2$

$Pr(H_{boy}|A) = \frac{1/3 * 0.6}{1/3 * 0.6 + 0.3 * 0.2 + 0.3 * 0.2} = 0.625$

So, required probability is approximately 62.5%