Probability of Full House

May 2024 ยท 2 minute read
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In a standard 5-card poker hand for 1 deck:

Calculate P(Full House)

Full House Hands

Full House

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 full houses = Possible ways to get 3 of a kind and a pair in one hand.

Possible 3 of a kind portion for full houses = 13 cards * 4 ways to choose 3 of a kind = 52 ways

Possible 1 pair portion for full houses = 12 possible choices <> 3 of a kind * 6 possible pairs = 72 ways

Total Possible full house combos (3 of a kind & 1 pair) = 52 * 72 = 3,744 ways

Probability of a full house  =  Possible full houses
  Total Hands

Reduce top and bottom by 624

GCF = Greatest Common Factor

P(Full House)  =  3,744
  2,598,960

GCF for 6 and 4165 = 624

P(Full House)  =  6
  4,165

Final Answer

Decimal probability = 0.0014405762


What is the Answer?

Decimal probability = 0.0014405762

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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