Probability question

Hello,

Can someone help me break down the math behind this hand?

I have: 9h Th

Flop: 7h 8h 5c

Opponent: 8d 8s

According to http://www.cardplayer.com/poker-tool...r/texas-holdem I have 42% of winning in an all-in.

How is it worked out?

So far, I've got to: I have 15 outs, 15 * 4 - (15 - 8) = 53% or in probability terms 0.53 chance.
Opponent has 35% to get his full house with 2 cards still to come, i.e. 0.35 chance.

I know that the probability of me winning is conditional, so I have to hit my outs, but also not manage to hit opponent's full house. Two cards to come (is this where the catch is? that I don't have 2 draws for my hand and 2 draws for opponent's hand, but 2 cards only for both hands, i.e. we share the draw?)
Why can't I just multiply the chances of me winning, 0.53 by the chances of opponent losing, 1-0.35 = 0.65? I only get to 0.3445, i.e. 34% not 42%.

Can anyone help my poor math?
thx

on this flop pokerstove (free equity calculator) says
8d8s = 57.879%
9h10h = 42.121%

if a blank comes on the turn

8d8s = 70.455%
9h10h = 29,545%
these are pokerstove calculations.

the 4 and 2 rule....
to get an approximate percentage of yours chances of improving
without knowing opponents hand
outs x 4 on the flop (2 cards to come)
outs x 2 on the turn (1 card to come)
eg;
in the hand in your post you have 14 outs to flush or str8 (2 of these will make a str8 flush)
so 14 x 4 gives 52% on flop (2 cards to come)
and 14 x2 gives 28% (1 card to come)
plenty of help here
check out the strategy section.
good luck m8

OP has already shown the equation for the '4 - 2' rule on flop (w 15outs) >
So far, I've got to: I have 15 outs, 15 * 4 - (15 - 8) = 53%
don't we typically use (15 - 6) here though, instead of ( -8)? > 51%

Not really sure how to work the math on this.
Obviously 4-2 rule is giving us a guesstimate of probability of hitting our hand.. but doesn't take into account villain's hand's improving.
(if we hit our draw on turn, villain has 20% chance of improving ftw... if we wiff, we're about 30% chance to win) < not sure how to do the math equations for this though

Opponents chance of filling up are 34.97 and then we have one 8 do 2.17 which is 37.14 which you don't look to have done in your calculation. If nobodies done it by tomorrow I'll go through the whole working out as I don't have much time tonight. Do you understand where we get the 34.97 from and the 2.17 from and just need to see the working out behind the 42% equity?

You have to do probabilites for each scenario seperately and once you have all possible equity combos combine them into one equity.

In fact I think you only needs 3 probabilities

We make straight flush

make straight, avoid 8 or FH

make flush avoid 8 or FH

It is obvious that you have already allowed for the 2 heart outs that make the flush and the straight, hence 15, the one that you missed is the 5h which completes villains boat, so 14 outs
It's also clear that you are familiar with 4&2 rule as you are using the more sophisticated (and accurate) version whereby you deduct all extra outs over 8 (the -(15-8) bit) after mutiplying
I always knew this as deduct #of outs over 9 (seems more accurate)... so

14*4=56%
56 - (14-9) =51% (I believe the actual is 51.2)

beyond this, you're right it is that the opponent could make a better hand, but I'm not sure how you got 35%, - board pairs, or 8c, or runner runner pair, but it will only affect your probability if it includes one of your outs (otherwise it's irrelevant) so I suspect the percentage is much lower than 35% (of villain improving over your improved hand)

@Poker Orifice
I tried doing -6 in my spreadsheet, and the errors are significantly higher!




@BlueNowhere
I am not sure where you get 2.17%; is it from the last 8 remaining in the deck? If so, my percentage of opponent filling up includes the quads (although I did fail to mention it).
Here's the calculations I've done using a program I wrote a while back because I couldn't calculate the probability using 2 cards to come because if the TURN card doesn't fill up, opponent has more outs on the river, so I just wrote a brute force of all combinations.
Anyways, that much is clear:

Three of a Kind to Full House or Quads
Flop-1-card-drawn: 15%
Turn-1-card-drawn: 22%
Flop-2-card-drawn: 33%


> You have to do probabilites for each scenario seperately and once you have all possible equity combos combine them into one equity.

I think I am going to give that a go because there are clearly several distinct scenarios. I also completely forgot about the straight flush (beating the fh and quads).

@Samango

Yep, thanks, I made another error with the number of outs, so yeah, I do have 14 outs, and yeah, I think 35% was a bit steep, it's actually 33% ignoring the straight flush, but with the straight flush possibility, this should be less.

Regarding Solomon's rule, I tested it in my spreadsheet, and your suggestion seems to be more accurate.

1) Tried deducting 9 instead of 8 (if over 8 outs).
More accurate than the standard Solomon's rule in mid range of outs.

2) Tried deducting 9 instead of 8 (if over 9 outs).
A little less accurate than both, and I don't think it's what you meant (I think it's #1 that you meant).

It's a little unclear what you mean when you say "deduct #of outs over 9", but instead subtract 9 from #outs.


---

I am going to try and work it out step by step. It's like going back to school *sigh*

I think it's just that Poker Orifice got his figures the wrong way round
I've never seen 6 used and I think he meant 15 outs - 9 (same as me) gives 6 ( I said 14-9 gives 5
I did actually mean deduct 9 where #of outs is over 9 ( so that's No2)
I know Harrington suggests 8 where # of outs is over 8 but for some reason I thought 9 was better.
In my example above 14*4-(14-9) =51% (actual 51.2%) seems better than 14*4-(14-8) = 50% but I haven't tested all possibilities so I'm not dismissing Harrington and Solomon's

In your original question you ask how your opponents probability is affecting you probability to win the hand (why you are 42% and not 51%), and this is where I think your estimate of 30-35% for your opponent is not helping your quest. Many of your opponents outs, for example 7d and a blank for you will improve your opponents hand but do not alter your probability
Only the opponents outs that also make your hand will affect your probability, Like 7d Jd. You improved but he improved better
This is going to be a much lower figure than 30%



when I say 'deduct number of outs over 9', I am referring to what you will deduct from the multiplied figure (14*4)
when you say 'subtract 9 from outs' you are referring to the interim step to arrive at that figure

so your 'subtract 9 from outs' is 14-9 = 5

5 is the 'number of outs over 9'
this is then deducted from the 14*4 step

We are talking about different stages in the same process

FLOP OUTS
---------

My outs flop -> turn: 14.
6 Hearts for a FLUSH, 3 Jacks and 3 Sixes for a STRAIGHT
2 Hearts for a STRAIGHT FLUSH
--
14/45 = 31.11%

Villain's outs flop -> turn: 31.
3 Fives, 3 Sevens, 1 Eight for a FH/QUADS.
24 BLANKs
--
31/45 = 68.88%

TURN OUTS
---------

--> 1) Turn card: BLANK, say, Kd
--------------------------------

My outs turn -> river: 13
5 Hearts for a FLUSH, 3 Jacks and 3 Sixes for a STRAIGHT
2 Hearts for a STRAIGHT FLUSH
--
13/44 = 29.54%
N.B. The blank that came on the turn is no longer our out for a flush.

Villain's outs turn -> river: 31
21 BLANKS (3 less blanks now: Kd @ flop, Kc and Ks are in the deck).
3 Fives, 3 Sevens, 1 Eight, 3 Kings (including the Kh) for a FH/QUADS.
--
31/44 = 70.45%

--> 2) Turn card: FH/QUADS
--------------------------

My outs turn -> river: 2 (for a STRAIGHT FLUSH)
2/44 = 4.54%

Villain's outs turn -> river: 42
42/44 = 95.45%

--> 3) Turn card: STRAIGHT, say, 6c
-----------------------------------

My outs turn -> river: 35
35/44 = 79.54%

Villain's outs turn -> river: 9
3 Fives, 3 Sevens, 1 Eight, 2 Sixes (6s and 6d) for a FH/QUADS.
9/44 = 20.45%

--> 4) Turn card: FLUSH, say, 3h
---------------------------------

My outs turn -> river: 34
34/44 = 77.27%

Villain's outs turn -> river: 10
3 Fives, 3 Sevens, 1 Eight, 3 Threes for a FH/QUADS.
10/44 = 22.72%

--> 4) Turn card: STRAIGHT FLUSH, say, Jh
-----------------------------------------

My outs turn -> river: 44, 100%
Villain's outs turn -> river: 0, 100%


My winning probability from flop to turn: 0.3111
My winning probability from turn to river scenario #1: 0.2954
My winning probability from turn to river scenario #2: 0.0454
My winning probability from turn to river scenario #3: 0.7954
My winning probability from turn to river scenario #4: 0.7727
My winning probability from turn to river scenario #5: 1.0000


And now I am stuck at putting it together.
Not exactly sure about #4 and #5 :|

This hand is actually from Gus Hansen's Every Hand Revealed book...he writes these percentages lightly, so I am glad I tried to work it out, so I am not even going to attempt to do this at a table.

OK Done...

FLOP -> TURN OUTS:

Villain's flop outs:
Blanks: 24/45 = 0.533333333
FH/Quads: 7/45 = 0.155555556
My flop outs:
Straight: 6/45 = 0.133333333
Flush : 6/45 = 0.133333333
Straight Flush: 2/45 = 0.044444444

TURN -> RIVER OUTS:

If turn = Blanks, my outs: 13/44 = 0.295454545 * 0.533333333 = 0.157575758

If turn = FH/QUADS, my outs: 2/44 = 0.045454545 * 0.155555556 = 0.007070707

If turn = STRAIGHT, my outs: 35/44 = 0.795454545 * 0.133333333 = 0.106060606

If turn = FLUSH, my outs: 34/44 = 0.772727273 * 0.133333333 = 0.103030303

If turn = STRAIGHT FLUSH, my outs: 44/44 = 1 * 0.044444444 = 0.044444444


Add them up:

0.157575758 + 0.007070707 + 0.106060606 + 0.103030303 + 0.044444444 = 0.418181818

So my conclusion is not to worry too much when you read/see percentages quoted in a book (the authors are showing off); the computations are clearly done on a computer (or what I just did), so if you can't quickly make the calculation, then probably you shouldn't be doing it.

@Samango
Check out the attachments.

Taking out 9 from the #outs if number of outs is over 9 is clearly a more precise calculation, so thank you for the info.
It is nearly bang on 11 through 17, no change 1 through 8; losing accuracy by 1 percent at 9 and 10, and again at 18 through 21.

Attached Images

Outs - Solomon (over 8).jpg (26.8 KB, 42 views)

Outs - Solomon (over 9).jpg (26.9 KB, 42 views)

Comparison - Precise.jpg (21.0 KB, 42 views)

Comparison - Rounded.jpg (21.0 KB, 42 views)

Cool, I knew there was some reason that I had been working with this version.
does it work just as well when using the 2 rule?

In your terminology you are referring to subtracting 9 or 8 (which is understandable, this is the aspect of the calculation that you are exploring here) but as I said before that is only a step in the process
I prefer 'subtract all outs over 9' which better describes what to do with your figure after you have used the basic 4 & 2 rule

this ^
actually the 15-6 was (15 - (15-9))

@Poker Orifice

> don't we typically use (15 - 6) here though, instead of ( -8)? > 51%
> actually the 15-6 was (15 - (15-9))

Sorry mate, but this still doesn't make sense. I get the 15-9, that's #outs minus 9 which is 6. But 15 - 6 makes no sense, unless you mean 15*4-6?

@Samango
> does it work just as well when using the 2 rule?

Well, there isn't a similar system for a Rule of 2 which is just muliplying the outs by 2. However, I did some searching, and there was a site mentioning an improvement by adding 1 if #outs is more than 6, however, I found that to be inaccurate still.

Here's what I've got (check out the attachment). The formula there was:
=IF(A2>9,(A2 * 2) + 2,IF(A2>3,(A2*2)+1,A2 * 2))
Basically means if number of outs is more than 3, add 1, if it's more than 9, add 2.
It's not perfect, but it irons things out a little. Towards 19-21 outs you're losing a whole 1 percent, but that's probably less important if you have shed load of outs; the calculation is a little more useful with less outs. No one needs such precise calculations anyway. You also have to bear in mind 1 fact when using Rule of 2, the percentage is slightly different on the flop and turn (because of less cards remaining on the turn), and that's nearly 1 percent towards the middle range of outs, so losing one percent here and there is OK because you're never going to remember the percentage for the flop and the turn.
With Solomon's Rule of 9, it is pretty straight forward, and because with the standard Rule of 4 the error is growing as you get more outs, Solomon's rule is actually necessary (without it you'll get 84% with 21 outs instead of 70%) whereas with this Rule of 2 modification, it's more work for less gain.
I think people will stick with the good old Rule of 2 and be done with it. It's a rough estimation and more than necessary. My pot odds calculations are a lot less accurate anyway.

Incidentally, I heard someone say at the casino the other day that 1 card = 2% (because of 50 cards in the deck), and I never thought of the connection with Rule of 2, how rough and nasty it is :-) But knowing this might help work out your outs % pre-flop, rough and nasty again though, e.g. hitting my set on the flop: 2 outs, 3 flop cards with each card being 2%, 2*3*2=12% to hit a set (after writing this I'll see if there's a dirty Solomon's rule for calculating outs preflop heh).

*sigh* enough of this....time to play some poker!

Attached Images

Rule of 2.jpg (36.1 KB, 20 views)

Yeah but not much work for a big gain in accuracy, I like it
(maybe not for calculating odds in a hand, but certainly for arguing odds with your mates lol)

prolly responded on this while tired.

What I typically use is:
ie.
15 outs (15 x 4) - (15 - 9) = (60 - 6)= 54%