2022 DHS J2 MYE P2 Q4

In this question you should state the parameters of any normal distributions you use.

In a high school with a large population, the test scores of the boys, \(B\), are normally distributed about a mean of \(55\) marks with standard deviation \(~11\) marks and the test scores of the girls, \(G\), are normally distributed about a mean of \(58\) marks with standard deviation \(8\) marks.

1. A boy and a girl are chosen at random. Find the probability that the difference between their scores is at least \(3\) marks.[3]
2. The probability that the mean score of four randomly chosen boys and five randomly chosen girls is less than \(m\) is more than \(0.85497\). Find the range of values of \(m\).[3]
3. A student whose test score exceeds \(70\) marks is classified as a smart student. Three boys and two girls are chosen at random. Find the probability that there are exactly two smart students among them.[4]