Probability coupon collection question - nth coupon is a new type? I'm just solving some probability problems in preparation for my exam, and I stumbled upon this one which I cannot tackle: Suppose that you continually collect coupons and that there are $m$ different types. Suppose also that each time a new coupon is obtained, it is a type $i$ coupon with probability $p_i, i = 1, \ldots ,m$. Suppose that you have just collected your $n$-th coupon. What is the probability that it is a new type? Hint: Condition on the type of this coupon. Any help would be appreciated, thank you.

Let $E_n$ be the event of interest (at the $n$-th extraction we get a new coupon type) and let $c_n=1 \cdots m$ be the type of the $n-$th coupon. Then

$$P(E_n) =\sum_{i=1}^m P(E_n | c_n=i) P(c_n=i) = \sum_{i=1}^m (1-p_i)^{n-1} p_i$$