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