A group of 4 friends are playing a card game using a standard deck of 52 cards. Each friend receives one card from the deck, and the next card is flipped up. Only one of the friend has a card that matches the suit of the flipped up card. What is the probability that only one person is able to have a suit that matches the suit of the card that's flipped up?

My work (it's most likely wrong):

Let's say that the turned up card is a hearts. There are 13 hearts in a deck of hearts.

13*39*38*37= 712,842

Only one of the cards is a hearts (13). The rest of the cards are not hearts (39, 38, 37).

The probability that one player has the matching suit -> 13/712,842.

The number is too small, and I feel that I did something wrong.


Three things:

  1. With the card being flipped one of the $13$ hearts, the player whose card is a heart as well must have one of the $12$ remaining hearts, not $13$. So you must use a factor of $12$ not $13$

  2. There are $4$ possible players whose card is a heart, so you need to have a factor of $4$ in there.

  3. Most importantly, you should not be dividing by the factors of $12$, $39$, $38$, and $37$: these are the possible target combinations that go in the numerator. The denominator are all the possible ways to get $4$ cards to the $4$ players, so there you get something like $51$, $50$, $49$, and $48$

OK, using these hints, try again!

  • $\begingroup$ @N.F.Taussig Right, thanks! :P $\endgroup$ – Bram28 Dec 3 '18 at 12:22
  • $\begingroup$ Since one card is placed on the table, there are $51$ cards left from which the four players can each select a card. $\endgroup$ – N. F. Taussig Dec 3 '18 at 12:27
  • $\begingroup$ Thanks! Would it be instead 51, 50, 49, and 48 since one of the cards is already used up? $\endgroup$ – Lauren Pablo Dec 3 '18 at 12:28
  • $\begingroup$ @laurenpablo exactly! $\endgroup$ – Bram28 Dec 3 '18 at 13:01

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