You have a standard pack of 52 playing cards. You then shuffle them and begin to draw out cards until you have three of a kind. What is the most likely number of cards drawn when this happens?

You then shuffle another pack of 52 playing cards into the pile. What happens to the expected number of cards now? (i.e. does it double / halve / stay the same?)

(In reply to Analytic Solution:Computer aided for computations by Charlie)

The computation of each term is somewhat of a hybrid of combinations and permutations, as one select card is in the final position, with all 52 possibilities considered there, but only the combinations are considered for the first n-1 positions without regard to the order they are drawn.

The same computation was done also using UBASIC for the added precision and the builtin COMBI function. The results were the same, but the program looks nicer in QuickBasic, without the line numbers. UBASIC helped in the debugging as the need for the restriction on p+s being less than or equal to 12 was found by the argument checking that UBASIC does on the COMBI function, that I did not build into the subroutines in the QuickBasic program.

Posted by Charlie on 2003-11-21 16:04:16