Question 1203217: Please show steps of how to solve the question.
Calculate the probability (as a percent) of the following 5-card poker hand:
A full house (three cards of one rank and two of another rank)
For example: 9♠, 9♥, 9♦, 2♣, 2♥
The Probability is: %
(round your percent answer to three decimal places)
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Answer: 0.144% (approximate)
Explanation
Use the nCr combination formula to determine how many five card hands are possible. Order does not matter.
n = 52 cards
r = 5 selections
n C r = (n!)/(r!(n-r)!)
52 C 5 = (52!)/(5!*(52-5)!)
52 C 5 = (52!)/(5!*47!)
52 C 5 = (52*51*50*49*48*47!)/(5!*47!)
52 C 5 = (52*51*50*49*48)/(5!)
52 C 5 = (52*51*50*49*48)/(5*4*3*2*1)
52 C 5 = 311875200/120
52 C 5 = 2598960
There are 2,598,960 ways to select five cards in any order.
Many calculators have a button labeled "nCr" to make such calculations quick and painless.
Here is one online calculator
https://www.gigacalculator.com/calculators/combination-calculator.php
The next step is to determine how many full house hands are possible.
Focus on the example trio 9♠, 9♥, 9♦
(ie 9 of spades, 9 of hearts, 9 of diamonds)
There's 1 card left out (9 of clubs). We do this to avoid getting 4 of a kind.
There are 4 ways to pick the unlucky card that gets left out.
Also there are 13 choices for the rank of this trio.
That gives 13*4 = 52 ways to form the trio.
For the duo (2♣, 2♥) we have 2 cards left out.
There are 4C2 = 6 ways to determine the suits for this duo. And there are 12 remaining ranks to pick from.
We have 6*12 = 72 ways to select the duo.
We found there are...- 52 ways to form the trio
- 72 ways to form the duo (after the trio is formed)
It yields 52*72 = 3744 different full house hands.
Again, order does not matter.
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In summary there are:- 3744 ways to get a full house
- 2,598,960 ways to select any five cards
Order does not matter.
Divide the values
3744/(2,598,960) = 0.0014405762305
This value is approximate.
Multiply by 100 to convert to a percent.
This moves the decimal point 2 spots to the right.
0.0014405762305 = 0.14405762305%
That rounds to 0.144% when rounding to 3 decimal places.
Further Reading:
https://en.wikipedia.org/wiki/Poker_probability
and
https://math.stackexchange.com/questions/808314/probability-of-getting-a-full-house
and
https://socratic.org/questions/592b200311ef6b2839d30feb
and
https://math.stackexchange.com/questions/4153143/how-many-ways-are-there-to-get-a-full-house
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