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.

    That probability is about 0.00037%

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

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


  4. 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?

  5. 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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