# Expected number of winners in this basket-like betting game?

Given $n$ players and $m$ baskets. Each player has random distinct label from the set $\{1, ..., n\}$. Each player $i$ selects a set of baskets $B(i)$ uniformly randomly with probability $p$. Then, each player $i$ puts a paper with his/her label in all the baskets in $B(i)$. A player $i$ win a basket $j \in B(i)$ if the paper he puts in $j$ is the smallest.

What is the expected number of winners ?

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Welcome to the site, @Ajed. Note that homework problems get special treatment here--you get hints to help you work though the problem for yourself, but not answers. These topics (& more) are discussed in our FAQ, which you may want to read. – gung Nov 1 '12 at 3:48
No No it is not a homework .. believe me. It just a question that came to my mind. – AJed Nov 1 '12 at 15:25
Sorry about that, @Ajed, it looked like HW to me. If you would prefer hints to help you think through the problem, leave it as is. If you just want someone to give you a complete solution, edit your question to remove the HW tag. – gung Nov 1 '12 at 15:33
no problem ! :) – AJed Nov 1 '12 at 18:14

Find the probability player $i$ wins a particular basket.
Use this to find the probability player $i$ wins at least one basket.