# The "tea bag" problem.

#### RWood

##### New member
I suspect this is not really an "advanced" probability question, but I'm not sure - haven't been near this stuff for decades.

The definition: I have a jar with 2*N tea bags in it (N>0 obviously). The beginning condition is that the teabags are joined in pairs - so there are N pairs. At each selection I select an item at random - initially that will be a pair of bags, in which case I tear one off and put the other back. On later turns I randomly select either a single bag, which would then be used, or a pair of joined bags (if there are any left), in which case I tear one off and proceed as above. What is the probability distribution - and hence the expectation - for D, the number of "drawings" required before there are no paired bags left in the jar?

It is clear that the values for D can range from N (by happening to always select paired bags) to 2*N-1.

I can see some ways of getting recursion equations, but I suspect that this problem has a simple answer resulting from a more general formulation. Any quick answers? Thanks.

#### CaptainBlack

##### Well-known member
I suspect this is not really an "advanced" probability question, but I'm not sure - haven't been near this stuff for decades.

The definition: I have a jar with 2*N tea bags in it (N>0 obviously). The beginning condition is that the teabags are joined in pairs - so there are N pairs. At each selection I select an item at random - initially that will be a pair of bags, in which case I tear one off and put the other back. On later turns I randomly select either a single bag, which would then be used, or a pair of joined bags (if there are any left), in which case I tear one off and proceed as above. What is the probability distribution - and hence the expectation - for D, the number of "drawings" required before there are no paired bags left in the jar?

It is clear that the values for D can range from N (by happening to always select paired bags) to 2*N-1.

I can see some ways of getting recursion equations, but I suspect that this problem has a simple answer resulting from a more general formulation. Any quick answers? Thanks.
I can't give you any help with this at present, I will have to think about it. However I can say I have seen this problem somewhere before, and vaguely recall it being connected with Herman Bondi (I suspect there was a note either in Mathematics Today or the Mathematical Gazette about it, but that is no help since my filling system makes it impossible to find even if I knew which and which year..).

CB

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#### RWood

##### New member
I can't give you any help with this at present, I will have to think about it. However I can say I have seen this problem somewhere before, and vaguely recall it being connected with Herman Bondi (I suspect there was a note either in Mathematics Today or the Mathematical Gazette about it, but that is no help since my filling system makes it impossible to find even if I knew which and which year..).

CB
Thank you for the update, will see what develops.