We consider the coupon collection problem, where each coupon is one of the types 1,…,s with probabilities given by a vector 𝒑. For specified numbers r1,…,rs, we are interested in finding 𝒑 that minimizes the expected time to obtain at least ri type-i coupons for all i=1,…,s. For example, for s=2, r1=1, and r2=r, we show that p1=(logr−log(logr))∕r is close to optimal.
"Optimality results for coupon collection." J. Appl. Probab. 53 (3) 930 - 937, September 2016.