## Random variables and probability distributions #1

A family health insurance policy pays the total of the first three claims in a year. If there is one claim during the year, the amount claimed is uniformly distributed between 100 and 500. If there are two claims in the year, the total amount claimed is uniformly distributed between 200 and 1000, and if there are three claims in the year, the total amount claimed is uniformly distributed between 5000 and 2000. The probabilities of 0, 1, 2, and 3 claims in the year are 0.5, 0.3, 0.1, and 0.1 respectively. Find the probability that the insurer pays at least 500 in total claims for the year.

I have the solution from the back of the book as follows, but I don't understand it:

T=total claims for the year, N=number of claims for the year.

$\displaystyle P[T\geq 500]=\sum^{3}_{0}P[(T\geq 500)\cap (N=k)]$
$\displaystyle =P[(T\geq 500)\cap P(N=2)]+P[(T\geq 500)\cap P(N=3)]$

(this is true since if there are 0 or 1 claim, then total must be $\displaystyle \leq 500$)

$\displaystyle P[(T\geq 500)\cap(N=2)=P[T\geq 500|N=2]*P[N=2]=\frac{1000-500}{1000-200}*0.1=0.0625$

$\displaystyle P[(T\geq 500)\cap(N=3)=P[T\geq 500|N=3]*P[N=3]=1*0.1=0.1$

Then $\displaystyle P[T\geq 500]=0.0625+0.1=0.1625$

In the emboldened part, why is it that if there is 1 claim, the total must be less than 500?

Also, how does $\displaystyle P[T\geq 500|N=2]=\frac{1000-500}{1000-200}$

Thanks