Four Of A Kind

Probability Level 2

In two decks of cards, what is the least amount of cards you must take to be guaranteed at least one four-of-a-kind?

The answer is 40.

1 solution

Rishabh Nishad
Mar 6, 2014

Any card drawn will be a A,2,3,4,5,6,7,8,9,10,J,Q, or K, so there are 13 possibilities each time a card is drawn.

The fastest way to draw a four of a kind is if the first four cards all have the same "value." The slowest way, which provides the solution, is to first draw 13 three of a kinds, and then one more card.

Since 13 x 3 + 1 = 40, if 40 cards are drawn it is guaranteed that those forty cards contain at least one four of a kind.

in the question it was written two deck of cards and the answer is of one deck of cards

yunus ali - 7 years, 2 months ago

Solve it using the pigeonhole principle. Instead of assigning the cards different values, imagine having one box for each different face of the deck. Since there are 13 different faces ( A,2,3,4,5,6,7,8,9,10,J,Q, and K), you will get 13 different boxes. Now, the question is how many cards (blank) do you have to put in these boxes to be sure that at least one of them contains 4 cards.

If we would have n boxes and wanted to make sure that at least one of these contained two cards, we would need 1 n+1 cards. If we want to make sure that n boxes contains 4 cards, we need 3 n+1.

Now we have 13 boxes representing a different face of the card (this is our n), so this means that we would have to use 13*3+1=40 cards to make sure that one of the boxes gets at least 4 cards placed in it. And this amount represents how many cards we have to take out of the two mixed decks to make sure we get at least four of the same kind.

Alma Andersson - 7 years, 2 months ago

Four Of A Kind

The answer is 40.

1 solution

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