Score Points: 1Solvers: 5

Take 13 cards of a suite (Ace, 2, 3, 4, ... , King).

The game is as follows: you pick two cards, and then the third.

If the third card is between the first two cards, you win.

What are the chances of your winning?

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Is the game any different if the first card needs to be in the middle of last two?

In essence, the question is the same as asking: what is the probability that out of any 3 cards drawn, the third card is in the middle in ascending order. Intuitively, there are 3 possible places the last card could be in ascending order, so the chance that it is in the middle should be 1/3. Mathematically, if 1 is the first card drawn, 2 is the second and 3 is the third, then the possible cases for arranging the three in ascending order are:
1,2,3
1,3,2
2,1,3
2,3,1
3,1,2
3,2,1
Out of these, the cases where the third card drawn is in between the first two are 1,3,2 and 2,3,1. Hence the probability of this happening is 2/6 or 1/3, which is 0.333