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

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.

• Is that supposed to be $1$ Million coins? It seems like because of our initial conditions the two pots are equally likely to recieve the first coin, so WLOG this equality should propagate through as you throw coins in the pots. So the smaller pot (smaller? maybe 1st pot) should be expected to contain half of $1$ million. Nov 20 '14 at 18:42
• @mapierce If I understand correctly the question and your answer, what you give is the expected value of the number of coins in the first or second pot and I think that we want the expected value of the minimum of the number of coins in the two pots. Nov 20 '14 at 18:56
• anon, I think you need to describe the problem a little more clearly. In particular, are $m$ and $n$ fixed values throughout the problem, or do you mean for them to refer to how many coins are in each pot, starting with $m=n=1$ and finishing when $m+n=1000000$? Also, when posting problems of this type, please include an account of your own attempt to solve them and a description of where you're stuck. This will help people give good answers at an appropriate level. Nov 20 '14 at 19:00
• @JeremyDaniel, my (tentative) interpretation is the OP is describing a "rich get richer" process in which the fuller pot attracts additional coins, in which case the expected final number in the smaller pot will be pretty small relative to the total number of coins. Nov 20 '14 at 19:05
• @BarryCipra: 1) m and n are no. of coins in the two pots at any given point of time. 2) At the time of finishing m + n = 1000002.
– anon
Nov 20 '14 at 19:13

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.