I had found this question.
A group of boys and girls know either French or Spanish. The number of boys and girls are in the ratio $1:4$. $30\%$ of the girls know Spanish and the rest of them know French. On the other hand, $50\%$ of the boys know Spanish and the rest of them know French. A student is chosen at random from the group of students who know Spanish. What is the probability that the chosen student is a girl?
My attempt was:
Let $P(G)$ and $P(B)$ be probability of choosing a girl and boy respectively. And $P(S)$ be probability of choosing someone who knows Spanish. Then,
$P(G|S)= \frac{P(S|G)*P(G)}{P(S|G)*P(G)+P(S|B)*P(B)}$
But the answer using this way is not in the option. Moreover (B) is the answer in their answer key.
(A) 2/7
(B) 12/17
(C) 20/41
(D) 8/13
Is something wrong with my approach? How can I approach it correctly?
Best Answer
The probability of choosing a girl is $\frac{4}{5}$ probability that a girl speaks Spanish is $\frac{4}{5}.\frac{3}{10}=\frac{12}{50}$ similarly probability of selecting a boy speaking Spanish is $\frac{1}{10}$ Thus using Bayes formula we get the answer as $\frac{12}{17}$