Have a look at this webpage.
In order to find the probability of dealing a full house in a 5 card poker hand, you use the folowing:
(13-choose-1)x(4-choose-3)x(12-choose-1)x(4-choose-2) / (52-choose-5)
But in order to find the probability of dealing 2 pairs, you use the this:
(13-choose-2)x(4-choose-2)x(4-choose-2)x(11-choose-1)(4-choose-1) / (52-choose-5)
What I don't understand is why doesn't the 2 pairs probability method work the same way as the full house method (choosing 1 from 13 then 1 from 12 rather than choosing 2 from the 13 at the start) and work as follows:
(13-choose-1)x(4-choose-2)x(12-choose-1)x(4-choose-2)x(11-choose-1)(4-choose-1) / (52-choose-5)
Hello, Phoebert!
I assume you understand the purpose of each of the factors.
There is a subtle difference in the two methods you mentioned.
In order to find the probability of dealing a Full House in a 5-card poker hand,
you use the following: .
Choose one of the 13 values for the Triple: ways.
Choose 3 of the 4 cards of that value: ways.
Choose one of the other 12 values for the Pair: ways.
Choose 2 of the 4 cards of that value: ways.
There are: . ways to get a Full House.
. . and divide by the number of 5-card hands:
. . to get the probability.
But in order to find the probability of dealing Two Pairs,
you use the this: .
Choose 2 of the 13 values for the Two Pairs: ways.
Choose 2 of the 4 cards of one value: ways.
Choose 2 of the 4 cards of the other value: ways.
Choose 1 of the other 11 values: way.
Choose 1 of the 4 cards of that value: way.
There are: . ways to get two Two Pairs.
. . divide by the number of 5-cards hands:
. . to get the probability.
What I don't understand is why doesn't the 2 pairs probability method work the same way as the Full House method
(choosing 1 from 13 then 1 from 12, rather than choosing 2 from the 13 at the start) and work as follows:
. .
Your method produces a number twice as large as necessary.
Here's the reason why.
The original method chooses 2 values from the 13 available values.
There are: possible choices for values.
We can list them if we like:
. .
Your method has: possible choices for values.
Your list looks like this:
. .
Your list considers to be different from .
That is "two Aces and two 2's" is not the same as "two 2's and two Aces."
And we know better . . .