Probability Of Making A Hand By The River

[VivaLaViking]

Assume a good flop comes for you, you see you have 17 outs to make a flush or filling an open ended straight, but you have still not made a hand. There are 47 unseen cards giving you ~ 36.2% (17/47) chance on the next card to make a hand. How can you infer your probability of making your hand by the river, post flop?

[IggyWH]

FYI, if you have a flush draw & open-ended straight draw, the max outs you can have is 15 since two of the cards that make your straight would also make your flush.

Example :

You have 8/images/graemlins/heart.gif 7/images/graemlins/heart.gif

Flop is :

6/images/graemlins/heart.gif A/images/graemlins/heart.gif 5/images/graemlins/club.gif

KQJT95432/images/graemlins/heart.gif will all give you a flush

9/images/graemlins/club.gif 9/images/graemlins/diamond.gif 9/images/graemlins/spade.gif 4/images/graemlins/club.gif 4/images/graemlins/diamond.gif 4/images/graemlins/spade.gif will give you a straight

The 9/images/graemlins/heart.gif & 4/images/graemlins/heart.gif would also give you a straight but are already counted as flush cards.

[BruceZ]

[ QUOTE ]
Assume a good flop comes for you, you see you have 17 outs to make a flush or filling an open ended straight, but you have still not made a hand. There are 47 unseen cards giving you ~ 36.2% (17/47) chance on the next card to make a hand. How can you infer your probability of making your hand by the river, post flop?

[/ QUOTE ]

Assuming 17 outs, even though you only have 15:

probability of hitting on the turn + probability of missing on the turn AND hitting on the river:

17/47 + (30/47 * 17/46) =~ 59.8%.


or 1 - probability of missing on both:

1 - (30/47)*(29/46) =~ 59.8%


or probability of hitting on the turn + probability of hitting on the river - probability of hitting on both:

17/47 + 17/47 - (17/47 * 16/46) =~ 59.8%


For 15 outs, substitute 15 for 17, 32 for 30, 31 for 29, and 14 for 16.

[KJL]

As the other poster siad, you only have 15 outs.

The way you figure it out is you subtarct the odds against making your hand from one.
1- {[(47-15)/47]*[(46-15/46]}= 54.1%

[VivaLaViking]

Yes, you are correct about the outs but what I really wanted to know is if you calculate your probability and have 2 chances is there a method to calculate the increase in the probabiliy because you have two shots. I tried unsucessfully to make the example as vague as possible. Just assume you have a 25% chance after the flop, is there a way to estimate the increase in probability because you have two more chances?

[Luzion]

Heres a very simple rule of thumb for approximating the probability you will hit your outs by the river from the flop. Use the 4x rule.

I found the real probability of getting the necessary outs by the river by the finding the complement of not hitting.


6outs would be 1 - (41/47)(40/46) = 24.1%
8outs would be 1 - (39/47)(38/46) = 31.5%
9outs would be 1 - (38/47)(37/46) = 35%
15outs would be 1 - (32/47)(31/46) = 54.1%

OR you could approximate these results by simplying multiplying the needed outs on the flop by 4.

6 x 4 = 24
8 x 4 = 32
9 x 4 = 36
15 x 4 = 60

This "4x rule" is accurate enough for the most common draws in hold-em and in omaha. Thought you might find it useful.

[Luzion]

[ QUOTE ]
Yes, you are correct about the outs but what I really wanted to know is if you calculate your probability and have 2 chances is there a method to calculate the increase in the probabiliy because you have two shots. I tried unsucessfully to make the example as vague as possible. Just assume you have a 25% chance after the flop, is there a way to estimate the increase in probability because you have two more chances?

[/ QUOTE ]

Yes, this is very simple to do. BruceZ, KJL, and I used this method in the post already.




Heres some real simple reasoning behind it.
1) You flip a coin. The probability of getting heads is 50%. Therefore the odds of NOT getting a heads is 50%.
2) You roll a dice. The probability of rolling a six would be 1/6. Therefore the odds of NOT rolling a six would therefore be 5/6.
3) A bag has 3green balls, and 1red ball. The probability of picking the red ball on your first try is 25% The probability of NOT picking the red ball is 75%.




See where I am getting at?

Now lets find the probability of NOT hitting your 9outs by the river. That would be (38/47) * (37/46) = 0.65.... Now you can reason that therefore the odds of HITTING one of the 9 outs at least once by the river is 0.35.

I hope I explained it as simply as possible... Basically find the probabillity of something happening, then subtract that result from 1, to find the probability of it NOT happening.



Heres a simple exercise so you can remember the logic. You hold a pocket pair before the flop. Can you tell me the probability you flop a set or better? (Hint: Find the probability of NOT hitting a set or better on the flop)

[VivaLaViking]

Thanks, you think like I do at a poker table. eg. Cloose enough. I would like to do some experimenting with it for some uncommon draws though, and most of them are in percents. Can this work in percents?

[Luzion]

[ QUOTE ]
Thanks, you think like I do at a poker table. eg. Cloose enough. I would like to do some experimenting with it for some uncommon draws though, and most of them are in percents. Can this work in percents?

[/ QUOTE ]

Im not exactly sure what you mean by "Can this work in percents?"

The 4x rule works great for the most common draws in Hold Em, which is 5, 6, 8, or 9 outs. It gets more and more inaccurate the more outs you try to figure out. 15 outs is really stretching the rule already since the probability of hitting 15 outs is 54% or so, while using the 4x rule you would get 60%.

If you are playing a game like Omaha Hi where its necessary to calculate your pot equity with as many as 20 outs, its best to learn how to do it by hand. I explained how to do this by finding the "complement" in my last post above this one.

[mosdef]

[ QUOTE ]
The 4x rule works great for the most common draws in Hold Em, which is 5, 6, 8, or 9 outs. It gets more and more inaccurate the more outs you try to figure out. 15 outs is really stretching the rule already since the probability of hitting 15 outs is 54% or so, while using the 4x rule you would get 60%.

[/ QUOTE ]

true, but often if the odds are there to chase a 60% draw, they'll be there for a 54% draw. so effectively, the "rule" will still work even if the probability estimate is off by as much as 6%

[VivaLaViking]

Luzion, Thanks for your help and now maybe I can pay you back a little.

Here are three series of number that should be easy to remember.

1 2 0 (9 - 11 Outs)
3 4 5 6 7 8 (12 - 17 Outs)
10 11 14 (18 - 20 Outs)

Using your 4x rule as before but with applying the proper series and subracting the proper series number your error up to and including 20 Outs will fall within a fraction of a percent.

Remember to shift series at 9, 12 and 18 outs. I watched the WPT this morning and tested it using a minimum of gray matter.

<font class="small">Code:</font><hr /><pre>
Outs % Subtract Resulting Error with Subtract
2 8.4% .4
3 12.5% .5
4 16.5% .5
5 20.4% .4
6 24.1% .1
7 27.8% .2
8 31.5% .5
.

9 35% 1 0
10 38.4% 2 .4
11 44.7% 0 .7
.

12 45% 3 0
13 48.1% 4 .1
14 51.1% 5 .1
15 54.1% 6 .1
16 57% 7 0
17 59.8% 8 .2
.
18 62.4% 10 .4
19 65% 11 0
20 66.3% 14 .3
</pre><hr />