Originally Posted by

**acc100jt** Suppose that there are $\displaystyle N$ distinct tyoes of coupons and each time one obtains a coupon it is, independent of prior selections, equally likely to be any one of the $\displaystyle N$ types. One random variable of interest is $\displaystyle T$, the number of coupons that needs to be collected until one obtains a complete set of at least one of each type. Rather than derive $\displaystyle P\{T=n\}$ directly, let us start by considering the probability that $\displaystyle T$ is greater than $\displaystyle n$. To do so, fix $\displaystyle n$ and define the events $\displaystyle A_{1}, A_{2}, ..., A_{N}$ as follows: $\displaystyle A_{j}$ is the event that no type $\displaystyle j$ coupon is contained among the first $\displaystyle n$, $\displaystyle j=1, ..., N$.

Hence, $\displaystyle P\{T>n\}=P\left(\bigcup^{N}_{j=1}A_{j}\right)$

I coulnd't understand the last equality, and why can't we derive $\displaystyle P\{T=n\}$ directly?

Appreciate those who help!!