Four of a Kind

Statistics Tutorial

Four of a Kind


This problem is an example of a statistical probability problem.

Sample Problem

If you were to choose 4 cards at random from a complete deck of cards (a 52 card deck), what is the probability that all of the cards chosen have the same value or in other words, you have a four of a kind? (ex. all fours)



To solve this problem, it must be understood that the probability of drawing one card is not independent (in other words dependent; however, AP Statistic exams prefer you use the words not independent) of the previous cards drawn. In other words, selecting a card from the deck will affect the probability of the choosing the next card.

1) Since we are trying to collect four of a kind, it doesn’t matter what the first card is, but the 3 cards after must have the same value. Since it doesn’t matter what the first card is, the probability of choosing the 1st card we want is a 1 or 100%.

2) To make things easier, let’s say the first card we chose was a 4 of diamonds. The next card we want to achieve a four of a kind must be a 4 of clubs, a 4 of spades, or a 4 of hearts. This gives us a total of 3 different cards left we need to draw from the deck. Since the deck originally had 52 cards and we removed the 4 of diamonds, there are now 51 cards left. There are 3 different fours left out of those 51 cards, so the probability of selecting two of a kind is 1 x (3/51)= 3/51= 5.88%

3) Let’s say the card we just chose was a 4 of clubs. There are now two fours left:the four of hearts and the four of spades. There are also now 50 cards remaining in the deck since we removed two cards. So the probability of drawing a third 4 is now (2/50). This makes the probability of drawing a three of a kind (3/51) x (2/50) = (6/2550) = .235%

4) Say we just selected a four of spades. That now leaves just one four left: the four of hearts. It also leaves 49 cards in the deck since we have now chosen 3 cards. So the probability of choosing the last 4 is a (1/49). So the probability of choosing a four of a kind is (6/2550) x (1/49) = (6/124,950) = .00480%

So the probability of choosing any four of a kind is .00480%

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