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Playing Cards
By Philip Ellison on Wednesday, January 02, 2002 - 04:20 pm:

If I had a pack of 52 cards and replaced one with a duplicate of another (ie. I now have 1 each of 50 "types" and and 2 of the other), and then dealt myself a hand of 13, what are the odds of my getting the double? I came up with 0.25, but I'm not very sure about this!

My reasoning:

1.No. of ways of arranging 52 items into 13 spaces:
52!/(13!×39!)

2.No. of ways of arranging 51 items into 13 spaces:
51!/(13!×38!)

3.As there are as many combinations (inc. duplicates) with 52 cards, no matter how many of each type you have, the number of "douhles" combinations must be:
52!/(13!×39!)-51!/(13!×38!)
which, when divided by 52!/(13!×39!) gives 0.25
Is the 3rd step valid, and if not what should I have done?
Also, if the pack is split between four people, what is the probability that at one of them will get the double (is it the same?)
Thanks

By Kerwin Hui on Wednesday, January 02, 2002 - 04:52 pm:

NO!

Suppose you replaced card A with card B. Then the event that you have a double B in your hand of 13 is the same as the event that you have both card A and card B in your hand had you not done the replacement. There are 50 choose 39 ways of dealing such a hand, and the total number of ways to deal a hand of 13 out of a pack of 52 is 52 choose 13. Hence the required probability is

[50!/(39!11!)]/[52!/(13!39!)]=13×12/(52×51)=1/17

A quicker way is to use the conditional probability

P(card A and B in hand)=P(card A)×P(card B|card A)
=(1/4)×(12/51)=1/17

Kerwin

By Graeme Mcrae on Wednesday, January 02, 2002 - 04:59 pm:

Your first question, which I assume means "what are the odds of getting both of the identical cards" is the same as asking of an ordinary deck,
"what are the odds of getting both the two and three of hearts?"

The probability of getting the two of hearts is 13/52, and then the probability of getting the three of hearts is 12/51. So the probability of getting both cards is (13/52)(12/51) = (1/4)(12/51) = (3/51) = 1/17, so the overall probability of one of the four players getting both cards is 1/17.

In your second question, the probability of one of the four people getting the two of hearts is 1. The probability of that same person getting the three of hearts is 12/51 = 4/17

By Yatir Halevi on Thursday, January 03, 2002 - 12:03 pm:

I fail to get why the probability of getting the two of hearts is 13/52.
Would you mind explaining it to me?

Yatir

By Philip Ellison on Thursday, January 03, 2002 - 12:44 pm:

13/52=1/4. Because all the cards are being dealt someone must receive the two of hearts, and as there are four people, the odds of it being you are 1 in 4 (you are receiving 13 out of 52 cards).
Thanks for the answers everyone... I really should have seen that it was that simple!

By Yatir Halevi on Thursday, January 03, 2002 - 03:25 pm:

Oh yeah.. ofcourse.
Thnx