Combinations Part III: Turn and River

Hey Guys,

I am going to talk about how to use combinations to know your probabilities for turn and river. I talk a lot about combinations, but they are an essential part for poker math.

Let's begin, I am sitting at a table and dealt AK suited of Clubs. To my excitement the flop is { 8h, 10c, Qc }. I am now on a flush draw, inside str8 draw, and I have two overs that can make a pair. I have a total of 18 outs that can make me a pair or better. For people who don't know what outs are. An out is how many cards I have left in the deck to make a hand. In this case a pair or better. Now, I have two cards left for the turn and river. What is the percentage that I will make a pair or better?

Well to start, how many cards have I personally seen. I have been shown 5 cards. The flop has 3 cards, and my hand has 2 cards. Out of a 52 deck I have NOT seen 47 cards.

Now, we start a combination formula. Since we are TAKING FROM the 47 unseen cards. The number 47 is our top part of the combination formula. Then we are selecting 2 cards AT TIME - turn & river. This makes the bottom part of our formula.
(47)
(2)

47 x 46 -----2162
______ = ________ = 1081
2 x 1---------2

1081 is the number of different combinations there are for the turn and river.

Now, for the turn, we now have the number 1081 we can go further into what our percentages are. Remember, I said we had 18 outs. We take 18 outs / 47 unseen cards for the turn.

Now for the river. We have to subtract our outs from the unseen turn cards. In this case 18 outs - 47 unseen cards = 29. We take this answer of 29 and divide it by 47 unseens card. Which would be (29 / 47). If we don't do this part of the process the river will make it easier to hit. We are still not done trying to figure out the river. Now, we divide our 18 outs by the 46 unseen cards on the river that are left in the deck. This looks like 18 / 46. The whole formula will look like this as below:

18 / 47 + (29 / 47) x (18 / 46)

Lets figure out this portion.

2.61 + (1.62 x 2.55)

Lets go further.

2.61 + 4.13

For the final parts to figuring this out. Remember, the number 1081? We take 2.61 / 1081, then 4.13 / 1081.

2.61 / 1081 = 414
4.13 / 1081 = 261

= Combinations: (Turn) 414 / 1081 + (River) 261 / 1081 = 675 / 1081
= Converted to Percentages: (Turn) 38.29% + (River) 24.14% = 62.43%


Okay, I hope this helps anyone out there. Remember, for those who go and use this on an online calculator. What we are assuming here: Is that none of our outs are taken and will not help anyone else to win. All of our outs will win, but we are already beat. We have not yet seen anyone elses cards.

If you go and use cardplayer's online calculator. You need to use atleast two players starting hands. Which means you will need to re-formulate my formula above. Since there are two or more SEEN cards.