Probability puzzle : Expected no. of coins in the smaller pot

Question

There are two pots of coins having size m & n. A new coin is thrown and goes to 1st pot with probability m/(m+n) and to 2nd pot with probability n/(m+n). We start with both pots of size 1 & 1 we throw 1M coins one by one. Find expected no. of coins in the smaller pot.

Answer 1

This is in effect a Pólya urn but with pots rather than colours.

Hints for the solution:

• Find the probability distributions of coins in the first pot after you have thrown $1,2,3,4$ coins.
• You should be able to see a pattern. Prove it by induction for more coins.
• Note that after you have thrown $1$ million coins, the pot with fewer coins has from $1$ to $500001$ coins and the one with more has $500001$ through to $1000001$ coins.
• Calculate the expected number of coins in the pot with fewer coins from the distribution you found. There is a minor point to take care of since you have thrown an even number of coins.

I will not spoil it by doing the calculation for you: it is easy and illuminating.