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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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Some Thoughts interesting problem... | Comment 1 of 39
I think it's difficult to analyze directly, because the conditional probabilities quickly grow quite complex... kind of like trying to analyze the game of black jack.

My thoughts would be to run this through computer simulations (this is a relatively easy problem to simulate) and analyze the results of performing one million trials of this.

Then we could determine the mean, median, mode, of the number of cards drawn.

BTW, it is not clear what this question is asking....

For instance, it may be that we can expect to get three of a kind (I'm making these figures up) before reaching the 22nd card (say... 50.0034% occur on or before the 22nd card), but it may be the most likely card to be the third of three-of-a-kind might be the 19th card pulled.

For which of these questions is Lewis after the answer? (Although if one performs the simulation, it should be trivial to answer both).

Lastly, a couple of observations...

  • One can't get three of a kind until you draw the 3rd card (obviously).
  • And you can't avoid getting a three of a kind on or before the 27th card.
    Edited on November 19, 2003, 3:53 pm
      Posted by SilverKnight on 2003-11-19 15:51:52
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