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Three of a Kind (Posted on 2003-11-19) Difficulty: 4 of 5
You have a standard pack of 52 playing cards. You then shuffle them and begin to draw out cards until you have three of a kind. What is the most likely number of cards drawn when this happens?

You then shuffle another pack of 52 playing cards into the pile. What happens to the expected number of cards now? (i.e. does it double / halve / stay the same?)

No Solution Yet Submitted by Lewis    
Rating: 4.3333 (9 votes)

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Solution A non-poker solution (computer program not used) | Comment 25 of 39 |
SilverKight wrote: "The problem, as you are interpreting it, is a much easier problem than interpreting it the way most of us are interpreting it. so, I suggest you solve the same problem that we are solving (and preferably do it without a computer)."

Fair enough. Here is the solution according to which "3-of-a-kind" just means "there are now three matching cards for the first time after n cards (n >= 3) have been drawn". This is for the 52-card deck only. Later I will edit this post for the 104-card case. I will just add that this version of the problem is immensely easier than the poker-rules version. So easy, in fact, that it is really not worthy to be on the flooble/perplexus website. :0)

There are 52*51*50=132600 ways to draw the first 3 cards from a 52 card deck. (SilverKnight says you need to divide by 3! to eliminate the duplicates. But as I conclusively demonstrated in "Some additional clarificatioins for SilverKnight", this is like "improving" an equation by dividing both sides by 1. You get the same result either way.) If these 3 cards are all aces, then there are 4*3*2=24 possibilities for these cards (e.g. spades/clubs/diamonds, hearts/clubs/spades...). So the odds of getting 3 aces in 3 draws are: 24/132600 = 0.0001809955

There are 52*51*50*49=6497400 ways to draw the first 4 cards from a deck of 52. If the 4rth card is an ace, there are 2+1=3 places 3 aces can be situated in 4 cards. For each of these, there are 48 possibilities for the non-ace. 24*3*48 = 3456 4-card draws containing 3 aces. The odds of this are 0.0005319051.

There are 52*51*50*49*48=311875200 ways to draw the first 5 cards from a 52 card deck. If the 5rth card is an ace, there are 3+2+1=6 places for the aces cards to be located among the 5 drawn cards. For each of these, there are 48*47 =2256 possibilities for the two non-aces. 24*5*2256 = 270720 5-card draws containing 3 aces. The odds of this are: 0.0008680395

Similarly, the odds...
For 6 cards: 0.0016991412
For 7 cards: 0.002493305
For 8 cards: 0.1097054206
For 9 cards: 0.2001292825
For 10 cards: 0.2513251454
For 11 cards: 0.3066765168
For 12 cards: 0.3656847354
For 13 cards: 0.4278511405
For 14 cards: 0.4926770708
For 15 cards: 0.5596638655
For 16 cards: 0.6283128636
For 17 cards: 0.698125404
For 18 cards: 0.7686028257

Looking at this data, it is my judgment that when we have 3 matching cards, in all probability we will have drawn 15 cards.


Edited on November 20, 2003, 6:06 pm
Edited on November 20, 2003, 6:09 pm
  Posted by Dan on 2003-11-20 18:03:41
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