Take a deck of cards labeled 1-60. Shuffle well and flip over the first 10 cards in a row from left to right.

Let x represent the number of cards in a row from the left that are in ascending order.

For example if the cards were 3 15 22 59 40 7 19 30 33 34 then x=4. The streak ends with the 59 because 40 is less.

Find a formula for p(x), the probability the first x cards are in ascending order and card x+1 is lower than card x, where x={1,...,10}

Generalize: For a deck of c cards labeled 1 to c and flipping over a row of n cards. Find a formula for p(x) where x={1,...,n}

I'll just give the general solution:

p(x)=x/(x+1)!

why?

suppose you asked for the probability that the first x+1 cards were in order.  That would be 1/(x+1)!.  Now, for each of those, take one of the first x cards and move it to the last position.  Those are all of the possibilities for first x in order and next one lower given some set of x+1 cards to start.  Thus, x/(x+1)!

Posted by Joel on 2006-10-10 12:46:53