High school probability problem

Question

I was reading on some problems as I wanted to learn probability distributions again from scratch.

However this problem I am not sure where to start as part a) mentions a score difference of 5 but not a specific score. Could anyone give me a clue on how to start a) and b)? It will be really helpful.

Answer 1

Guide:

For part $(a)$, Let $B$ be the normal distribution for boys and $G$ for the distributin for $G$.

You want to compute $P(|B-G|<5) = P (-5<B-G<5) $

Find out the distribution of $B-G$, then you can use it to compute the desired probability.

If you understand part $(a)$, you can do part $(b)$ too.

Answer 2

You are adding and subtracting normal distributions here. If $A\sim\mathcal N(\mu_1,\sigma_1^2)$ and $B\sim\mathcal N(\mu_2,\sigma_2^2)$ are random variables, then $A+B\sim\mathcal(\mu_1+\mu_2,\sigma_1^2+\sigma_2^2)$. This is a nice property pretty much unique to the normal distribution.

Let $X$ be a random variable representing a boy's score and $Y$ the one for the girls. For the first, compute $P(|X-Y|<5)$. For the second, compute $P(X<\frac34Y)$.