Professor X smokes a pipe. He carries two identical matchboxes, originally containing 20 matches each. When he lights his pipe, he chooses a matchbox at random and lights his pipe with one match and discards the used match.
There will eventually arise an occasion when he first selects a matchbox with only one match in it. At this point, what is the expected number of matches in the other box?
(In reply to re(2): Corrected Solution after Charlie's comments
by Dan Rosen)
The probability P20 should answer a completely different scenario, namely :
"The professor has drawn randomly an unknown number of matches, stopping when either one of the boxes is left with one match. What is the probability that when that occurred, 20 matches were left in the other box."
However P20 is but one term in a summation, or rather, a factor in but one term in a summation, the other factor being the number 20 itself, to convert probability into expectation. It is only the one expectation component based on a twenty pick condition; the other length conditions have their own probability. This one is the same as getting 20 heads in a row when tossing coins.
Posted by Charlie
on 2010-07-01 20:11:51