## Monday, November 16, 2009

### Playing With the Queen of Hearts

Two identical packs of 52 cards A and B are shuffled thoroughly. One card is picked from A and shuffled with B. The top card from pack A is turned up. If this is the Queen of Hearts, what are the chances that the top card in B will be the King of Hearts?

1. Here's an attempt...

Your chances are 1/53 if the King of Hearts was not the card you moved from pack A to pack B, and 2/53 if it is the card you moved. The odds of the King of Hearts being the card you moved over are 1/51 (since you eliminated the Queen of Hearts). So I'd say the chances are:

(1/53 * 50/51) + (2/53 * 1/51) = .01924ish

2. Andy has explained with very well, and I reached the same equation.

SO I guess that is the answer.
.01924....

www.guessthelogo.blogspot.com

3. is it.. 52 / (51 * 53)

4. This is a little tricky. The answer is 1/53 + (1/53 * (1/51)) which reduces to 52/2703 or approximately 1.92378838%.

The trick is this. If the top card of A is the Queen of Hearts then the chances that we picked the King of Hearts out of A and put it into B is 1/51. The chances of picking the King of Hearts from the original B deck are 1/53 (53 because we added the card from A) and the chances that we pick the a King of Hearts from A shuffled into B is 1/51 * 1/53. So the total chances are 1/53 + 1/53 * 1/51.

The big trick is the Queen of Hearts. If the question said that you turned up some random card from A then the answer would be 1/53 + 1/53 * 1/52. But since we know it is the Queen then we know that the card we took from A is not the Queen so it is one of the 51 other cards in the deck.

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