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(5): Corrected Solution after Charlie's comments by Dan Rosen)

I assume you're referring to such a situation as:

 19 18 18 17 16 15 15 15 15 15 15 14 14 13 12 12 11 10  9  8  7  7  7  6  5  4  4  4  3  3  3  2  1  1  1<
 20 20 19 19 19 19 18 17 16 15 14 14 13 13 13 12 12 12 12 12 12 11 10 10 10 10  9  8  8  7  6  6  6  5  4
1  1  2  1  1  1  2  2  2  2  2  1  1  1  1  2  1  1  1  1  1  2  2  1  1  1  2  2  1  2  2  1  1  2  2

where I've added a row indicating which box the professor has chosen, explaining the transitions shown in the lines above.

You are saying this trial does not count, as he did not select the box containing the single match at the first opportunity.  But the statement of the problem does not require that he select it at the first opportunity, but rather asks concerning the time when he does eventually choose the single-match box.

This is evidenced by the fact that the problem states that when the professor acts in accourdance with the first paragraph, that
"There will eventually arise an occasion when he first selects a matchbox with only one match in it. " It doesn't say, there may arise such an occasion, or even that there eventually did occur such an occasion, but, rather, there definitely will, whenever the professor acts in the manner described in the first paragraph. The "first" is not referring to the first occurrence of a solitary match, but rather to the first time he selects it.  And such a time does in fact exist in the sample trial shown above.

Further, if, as you suggest, those instances where the choice of the single-match box was not taken at the first opportunity were excluded, the expected number of matches in the unselected box at the end would increase, as the successive reductions in the over-1 box would not have taken place.  A simulation of what you are saying results in the expected value going up to 5.9, even farther from your calculated values.

Posted by Charlie on 2010-07-21 13:44:44