Probability of 3 of a kind

Publish date: 2024-06-05
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In a standard 5-card poker hand for 1 deck:

Calculate P(3 of a kind)

3 of a kind Hands

3 of a kind

Calculate Total 5 Card hands

Choose 5 cards from 52 cards

Total Hands  =  52!
  (52-5)! * 5!

Total Hands  =  52!
  47! * 5!

Total Hands  =  (52 * 51 * 50 * 49 * 48) * 47!
  47! * (5 * 4 * 3 * 2 * 1)

Cancelling the 47!, we get:
Total Hands  =  311,875,200
  120

Total Hands = 2,598,960

Build Probability

Possible 3 of a kinds = 13 cards to choose * 4 ways to draw that card = 52

Possible 3 of a kinds = 52 ways to draw the 3 of a kind * 48 4th cards = 2,496

Possible 3 of a kinds = 2,496 * 44 5th cards not matching the 4th card = 109,824

Possible 3 of a kinds = 109,824 possibilities/2 ways to draw cards 4 & 5

Possible 3 of a kinds = 109,824/2

Possible 3 of a kinds = 54912

Probability of a 3 of a kind  =  Possible 3 of a kind hands
  Total Hands

Reduce top and bottom by 624

GCF = Greatest Common Factor

P(3 of a kind)  =  54,912
  2,598,960

GCF for 88 and 4165 = 624

P(3 of a kind)  =  88
  4,165

Final Answer

Decimal probability = 0.0211284514

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What is the Answer?

Decimal probability = 0.0211284514

How does the 5 Card Poker Hand Calculator work?

Free 5 Card Poker Hand Calculator - Calculates and details probabilities of the 10 different types of poker hands given 1 player and 1 deck of cards.
This calculator has 1 input.

What 1 formula is used for the 5 Card Poker Hand Calculator?

Total Possible 5 Card Hands = 2,598,960

For more math formulas, check out our Formula Dossier

What 4 concepts are covered in the 5 Card Poker Hand Calculator?

5 card poker handcombinationa mathematical technique that determines the number of possible arrangements in a collection of items where the order of the selection does not matter
nPr = n!/r!(n - r)!factorialThe product of an integer and all the integers below itprobabilitythe likelihood of an event happening. This value is always between 0 and 1.
P(Event Happening) = Number of Ways the Even Can Happen / Total Number of Outcomes

Example calculations for the 5 Card Poker Hand Calculator

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