Expected number of draws

In an urn there are $n$ balls each with weight $w_i$, drawn one at a time without replacement. At any point of drawing, the probability of a ball being drawn is proportional to its weight. What's the expected number of draws before we draw ball 1?

Solution: Coupon drawing

Original problem here: http://dharmath.thehendrata.com/2010/01/12/coupon-drawing/

A container holds $N$ coupons. You draw successive coupons from the container, observe the coupon drawn, and then replace the coupon. What is the expected number of draws needed until all $N$ coupons have been seen at least once?

Challenge: what is the probability that the process ends after exactly $M$ draws?

Coupon drawing

Credit for this problem goes to Sander Parawira

A container holds $N$ coupons. You draw successive coupons from the container, observe the coupon drawn, and then replace the coupon. What is the expected number of draws needed until all $N$ coupons have been seen at least once?

Solution here: http://dharmath.thehendrata.com/2010/01/12/solution-coupon-drawing/