## Wednesday, July 26, 2006

### What are the odds?

Four cards are dealt from the top of a well shuffled deck. What are the odds that they are four aces?

1. I would assume it's
4/52 * 3/51 * 2/50 * 1/49

the logic being that you can draw any of the 4 aces first, out of 52 possible cards, then any of the three remaining aces from the 51 remaining cards, etc etc.

2. Pretty good if the deck is all aces...

3. Thanks Jon... I really needed the laugh! :-)

4. "Pretty good if the deck is all aces..."

Nice

5. The probability is: (4/52)*(3/51)*(2/50)*(1/49) = 1/270,725

Which makes the odds 270,724 to 1.

Maybe the better question, what are the odds the deck is well-shuffled?

6. There are two ways (that I can think of) to solve this one. First, the easy way is to use the formula N choose r. Since the number of distinct groups of four cards is 52 choose 4 = 270,725 and only one of those is four aces, the odds are 1 in 270,725.

Or you can do it this way. The odds of pulling an ace on the first draw is 4/52. The odds of pulling an ace, now that we have an ace on the second draw is 3/51. The odds on the third is 2/50 and the fourth is 1/49. Put them together and you get 4/52 * 3/51 * 2/50 * 1/49 = 1/270,725.

As to the odds of the deck being well shuffled, I'm not that good at statistics! ;-)

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