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acc100jt · #1 July 1st, 2009, 07:36 PM

Suppose that there are $N$ 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 $N$ types. One random variable of interest is $T$ , 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 $P\{T=n\}$ directly, let us start by considering the probability that $T$ is greater than $n$ . To do so, fix $n$ and define the events $A_{1}, A_{2}, ..., A_{N}$ as follows: $A_{j}$ is the event that no type $j$ coupon is contained among the first $n$ , $j=1, ..., N$ .
Hence, $P\{T>n\}=P\left(\bigcup^{N}_{j=1}A_{j}\right)$

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

Appreciate those who help!!