Suppose that there are 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 types. One random variable of interest is , 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 directly, let us start by considering the probability that is greater than . To do so, fix and define the events as follows: is the event that no type coupon is contained among the first , .
I coulnd't understand the last equality, and why can't we derive directly?
Appreciate those who help!!