At a high school anniversary program, students from only classes 9 and 10 participated. 65% of participating students are boys. If a boy is picked, the probability of the boy being picked to be a class 10 student is 0.3. What is the probability of picking a class 9 girl student if the ratio of class 9 students to 10 is 3:2?

It is given that 65% of the total are boys. So, the probability of picking a boy is 0.65.

We are given that if a boy is picked, the probability of the boy being a class 10 student is 0.3.

Using Bayes theorem $$P\left(10\left|B\right|\right)=\frac{P\left(10\ \bigcap\ B\right)}{P\left(B\right)}$$

$$0.3=\frac{P\left(10\ \bigcap\ B\right)}{0.65}$$

$$P\left(10\ \bigcap\ B\right)=0.3\times0.65\ =\ 0.195$$

$$P\left(9\ \bigcap\ B\right)=0.65-0.195\ =\ 0.455$$

We are told that class 9 and 10 students are in 3:2. So the probability of picking class 9 students is 0.6.

We know that $$P\left(9\ \bigcap\ B\right) =\ 0.455$$. From this $$P\left(9\ \bigcap\ G\right)=0.6-0.455\ =\ 0.145$$