Probability

[Luca]

Provided you have been dealt 2 cards to make the hand, what are the probabilities of making the following hands by the river?

Royal Flush
Straight Flush
4 of a kind

As an example I'm asking if you have been deal Kh Qh whats the chance of making a royal flush by the river? What is the formula you use to work this out?

Much appreciated if anyone can help with this.

[pete shaw]

You will get more/better responses if you post in the probability(or math)forum. [img]/images/graemlins/wink.gif[/img]

[KJL]

It depends on what your starting hands are. But I will tell you the odds for the KhQh.
Royal Flush: [1*1*1*C(47,2)]/C(50,5)=~.05% or ~1960-1
Straight Flush: same as above
Quads: [2*C(3,3)*C(47,2)]/C(50,5)=~.1% or about 980-1
So overall it is about a .2% chance to make quads a staright flush or quads by the river.

Note: These odds are somewhat incomplete. For the staright flush or royal flush I am only including the times when you use both of your hole cards. For quads I am only including times when there are three queens or kings on the board. In other words, if the board read: 2 [img]/images/graemlins/heart.gif[/img]2 [img]/images/graemlins/club.gif[/img]2 [img]/images/graemlins/diamond.gif[/img]2 [img]/images/graemlins/spade.gif[/img]J [img]/images/graemlins/heart.gif[/img], you would technically have quads, but I did not include this in my calculations.

[tpir90036]

1) try the probability forum
2) the math used is called combanitorics if you want the exact answer. you are counting up all of the possible outcomes.
3) see here for some articles on the subject:
http://www.math.sfu.ca/~alspach/pokerdigest.html

[AaronBrown]

Given any two pocket cards, there are 2,118,760 possible boards.

For a royal flush you need three specific cards to make the hand, assuming you start with two. The other two board cards can be any combination of the remaining 47 cards, there are 47*46/2 = 1,081 of them. There are three more boards that give you a royal flush on the board. So the odds are 1,084/2,118,760 = 0.05% (1 in 1,955).

The best two cards for getting a straight flush are suited connectors from J/10 to 5/4. These hands allow four different straight flushes. Other hands like A/K or J/7 allow only one that uses both cards. The answer is approximately equal to 0.05% times the number of straight flushes that can form, the exact answer depends on the two cards.

If you have a pair in your hand, you need two specific cards on board to get quads, with 48*47*46/(3*2) = 17,296 ways to get the other three cards. In addition, there are 12*46 = 552 ways to get four of a kind on the board. So you have 17,848/2,118,760 = 0.84% (1 in 119) chance of getting four of a kind.