Concerning probability in mathematics

[pigtimmy0722]

If we consider the hierarchy of hands, where a Full House with Aces is the highest and a Full House with Twos is the lowest, what is the probability of obtaining a Full House and winning with it out of a standard deck of 52 cards?

In a standard deck of 52 cards, there are 13 ranks (A, 2, 3, ..., 10, J, Q, K), and for each rank, there are 4 suits. To calculate the probability of obtaining a Full House and winning with it, we'll break it down step by step.

Choose the rank for the triplet (3 of a kind):
There are 13 ranks to choose from. So, you can choose one rank out of 13:
(13/1)
( 1/13).

Choose 3 suits for the triplet:
For each rank, there are 4 suits, so you need to choose 3 suits out of 4:
(4/3)
( 3/4).

Choose the rank for the pair:
Once you've chosen the rank for the triplet, you have 12 remaining ranks to choose from for the pair:
(12/1)
( 1/12).

Choose 2 suits for the pair:
For each rank, there are 4 suits, so you need to choose 2 suits out of 4:
(4/2)
( 2/4).

Total ways to get a Full House:
Multiply the above choices to get the total ways to get a Full House.

Total possible hands:
This is simply choosing 5 cards out of 52:
(52/5)
( 5/52).

Calculate the probability:
The probability of getting a Full House is the total ways to get a Full House divided by the total possible hands.

Putting it all together:

Probability of Full House
=
(13/1)×
(4/3)×(12/1)×(4/2)
(52/5)
Probability of Full House=
( 5/52)
( 1/13)×( 3/4)×( 1/12)×( 2/4)
​
references : Mathematics for homeschoolers