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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?
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.
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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I would assume it's
ReplyDelete4/52 * 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%
Pretty good if the deck is all aces...
ReplyDeleteThanks Jon... I really needed the laugh! :-)
ReplyDelete"Pretty good if the deck is all aces..."
ReplyDeleteNice
The probability is: (4/52)*(3/51)*(2/50)*(1/49) = 1/270,725
ReplyDeleteWhich makes the odds 270,724 to 1.
Maybe the better question, what are the odds the deck is well-shuffled?
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.
ReplyDeleteOr 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! ;-)