Question about Outs Computation in Advanced Texas Holdem

[Player_X]

All,

When I look at the probability section of Advanced Texas Holdem, Appendix A, page 309 it says that for 1 out the percentage is 4.4 and for two outs the percentage is 8.4. When I compute them I get 4.3 and 8.6.

Can someone please tell me what I am doing wrong?

One out:

1 / 47 = .0021276
1/ 46 = .0021739
.0021276 + .0021739 = .0043015 = .0043015 * 100 = 4.3015

Two outs:

2 / 47 = .0042553
2/ 46 = .0043478
.0042553 + .0043478 = .0086031 = .0086031 * 100 = 8.6031

Thanks in Advance,

Player_X

[Ed Miller]

Can someone please tell me what I am doing wrong?

Your method is not correct, but it will yield accurate approximations when the number of outs is small.

The problem is that there is an OR in your probabilities. You are trying to calculate how often A OR B happens (you make on the turn OR you make on the river). You cannot simply add the probabilities together because sometimes A AND B will happen.

The exact way to calculate is to find the probability of A OR B is to find the probability of (NOT A) AND (NOT B) and subtract that from 1. That is,

Prob of one out = 1 - (46/47)(45/46)

Prob of two outs = 1 - (45/47)(44/46)

[uuDevil]

Despite my shoddy math skills, I'll give it a shot.

You cannot add the probabilities the way you have done it because the term for making your hand on the river doesn't account for the fact that you have to have missed on the turn in order to be looking to make it on the river. So the river term should be

P(missed on turn)*P(hit on river)

then for the first case

P(making your hand)= P(hit on turn)+P(missed on turn)*P(hit on river)=

(1/47)+(46/47)*(1/46)= 0.04255 or about 4.26%

and for the second case

(2/47)+(45/47)*(2/46)=0.08418 or about 8.42%

The first number doesn't agree with your quoted value, so as a check I calculate it as

1-(the odds of NOT making your hand):

1-(46/47)(45/46)=0.04255, same as above.

Lastly, if I do a PokerCalculator (Monte Carlo) simulation, w/ JJ on a Jxx board, I get 42397 quads in 1,000,000 hands, or about 4.24%, close to the calculated value.

So there is still a discrepancy in the first case. I don't know why... /images/graemlins/blush.gif

Edit: Damn it. That Miller guy is everywhere these days! /images/graemlins/grin.gif

[Player_X]

Thanks Ed, and uuDevil,

So using Ed’s formula:

1 – (( 46/47 ) * ( 45/46 ))
1 - (.9787234) * (.9782609)
1 - .9574468

= .0425532 * 100 = 4.25532 = 4.3 <> 4.4

Am I still doing something wrong? This is supposed to be 4.4?

Thanks in Advance,

Player_X

[BugsBunny]

No I still get 4.3 I also get 20.4% for 5 outs rather than the 20.3% published. All the others I agree with. I think the book is in error on these 2.

[uuDevil]

Your number agrees with mine. The chart appears to be wrong. My copy also says 1 out is 4.4%. It is a small error, perhaps a typo. It isn't of any practical significance.

Nice catch, though.