I need an equation for what is the Probability of getting a full house in a 5 card draw poker game.?

Question

Noting that a full house is 3 of one card and 2 of another.

Answer 1

This is how I normally do it.

There are a total of C(52,5) hands.

To draw a full house,

There are 13 possible card you can draw for the first one. And you must draw three of the same card from 4 suits.
The second card can be any of the 12 cards you haven't drawn and you need two

So you have 13*C(4,3) * 12*C(4,2)

probability is [13*C(4,3)*12*C(4,2)] / C(52,5)




Alternatively, there is another way to think about it.

For the first card, you can choose anything so you have 52/52 cards to choose from.

The second one is the same card as the first so you have 3/51 cards that work. The third one you have 2/50 that work.

The second card is the 48/49 not used. The last card is the same type of card so you have 3/48 that work

You have 1*3/51 * 2/50 * 48/49 * 3/48 * C(5,2) --->there are C(5,2) ways to arrange the cards. You do not have to draw them out in any particular order.

Answer 2

13 choices for which card you have 3 of.
4C3 = 4 ways to draw 3 of them
12 remaining choices for which card you have 2 of
4C2 = 6 ways to draw 2 of them.

13(4)(12)6 = 3744 full houses.

divide by 52C5 = 2,598,960 poker hands.

probability is 0.00144

Answer 3

This hand has the pattern AAABB where A and B are from distinct kinds. The number of such hands is (13-choose-1)(4-choose-3)(12-choose-1)(4... The probability is 0.001441.

Answer 4

http://en.wikipedia.org/wiki/Poker_probability

this page has the probabilities for all poker hands.