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The Mathematics of Poker (1 Viewer)

chkmte

chkmte

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I'm going to present an example hand, followed by three questions. Please provide only number/fact-based answers and be as succint as possible. Explain it to me like we're in third-grade.

Here is the situation:

Player A holds Ac / As, Player B holds 9h / 10d. The flop comes 8c / Jd / 2h. There is $100 in the pot. Player B will fold his open-ended straight draw if faced with a bet which exceeds his immediate odds. (Note: Implied odds are not part of the discussion)
  1. What is the least amount that Player A can bet to force Player B to fold?
  2. How did you determine the answer to the first question?
  3. Is there a shortcut to determine said bet?
As many of you are aware, there is a shortcut which helps one determine their "outs". It's called the rule of 2 & 4. What I'd like to do is to come up with a similar (bet-sizing) shorcut, using only information which is readily available.

Thanks in advance, I look forward to your comments.
 
Player B has seen 5 cards total (his and the flop). Of the 47 remaining cards, 8 are outs to an almost certain win (all 7s and all Qs) and 39 are not. Thus his odds of hitting an out ON THE NEXT CARD (his immediate odds) is 39-8. Player A must be more than $25.80 (e.g., $26) to make Player B fold.

Not sure this is a shortcut but I think the math would be [(Pot Size)+(Player A Bet)]/(Player A Bet) would be odds that Player B gets by calling, which is also the same as (Pot Size)/(Player A Bet)+(Player A Bet)/(Player A Bet)or [(Pot Size)/(Player A Bet)]+1. Solving for Player A Bet you get:

Player A Bet > (Pot size)/(odds-1) to force B out. In this case >100/[(39/8)-1] or >100/(31/8) or >$25.80

But remember that this is only based on the odds of him hitting on the turn. The odds of him hitting on the river can be fairly estimated by dividing his turn odds by 2.
 
Using the rule of 2 shortcut (rough odds with one card to come) there are 8 outs, so 16% chance to make the straight.

Odds are then 84:16, or roughly rounded a little over 5:1. So you need to bet in a way that gives him worse than 5:1.

Any bet over $25 will do this.

You bet $25, pot is then $125, he needs to call $25 to win $125, he's getting 5:1.

Rule of thumb at the table: when in doubt, bet 2/3 pot. This gives poor odds for most draws and good value against 2nd best hand that will pay off.
 
Andrew Marks said:
Player B has seen 5 cards total (his and the flop). Of the 47 remaining cards, 8 are outs to an almost certain win (all 7s and all Qs) and 39 are not. Thus his odds of hitting an out ON THE NEXT CARD (his immediate odds) is 39-8. Player A must be more than $25.80 (e.g., $26) to make Player B fold.

Not sure this is a shortcut but I think the math would be [(Pot Size)+(Player A Bet)]/(Player A Bet) would be odds that Player B gets by calling, which is also the same as (Pot Size)/(Player A Bet)+(Player A Bet)/(Player A Bet)or [(Pot Size)/(Player A Bet)]+1. Solving for Player A Bet you get:

Player A Bet > (Pot size)/(odds-1) to force B out. In this case >100/[(39/8)-1] or >100/(31/8) or >$25.80

But remember that this is only based on the odds of him hitting on the turn. The odds of him hitting on the river can be fairly estimated by dividing his turn odds by 2.
Click to expand...

Wouldnt 8 outs = a 32% probability of hitting your card on the turn (8X4)?
 
mummel said:
Wouldnt 8 outs = a 32% probability of hitting your card on the turn (8X4)?
Click to expand...

32% is the probability of hitting our card with two streets to come. We can do that only if one of us is all in.

Edit: If we are looking at two streets, we pick up additional outs: two 9s, two 10s, and one 9 plus one 10. For the turn, any 9 or 10 is 6/45. If we get one of these cards, any of the remaining 9s and 10s is 5/44. Multiplying these gives us a probability of hitting two pair or trips is 1.52%. This is less than the value of a single out, showing that the odds of improving from no pair to two pair or trips are very, very low. In this case, going to the river is approximately 32% + 1.5% or 33.5%.
 
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To those who participated in the thread, thank you. I appreciate you taking the time to chime in.

- - -

After doing some MS Excel wizardry, I may have discovered a solution. While I'm not sure my "shortcut" is original, I certainly can't find it anywhere on the internet. It could be that I've discovered something unique - I really don't know. Without further ado, here's what I've come up with to help determine proper 'bet sizing' on the fly.

The rule of 3 & 6: Take the (maximum) # of outs (you suspect your opponent has) x 3. Make a bet equal to that number/percentage of the pot.

For example:
You suspect your opponent has an open-ended straight draw (8 outs). There is $50 in the pot. Take 8 x 3 = 24...bet 24% (or 25%) of the pot - which is about $13. In this scenario you're giving your opponent pot odds of 19.4% while true odds are 19.1%. So, using this rule - you're going to be very accurate. Of course, you can just double that number if it's a situation that will run "flop-to-river".

Another example:
You put your opponent on a flush-draw with an over-card (11 outs). There is $84 in the pot. Take 11 x 3 = 33...bet 33% of the pot - which is about $56. In this scenario you're giving your opponent pot odds of 24.8% while true odds are 23.4%.

Obviously, you can be more or less aggressive with your bet selection - but this gives you a good baseline to work from (I think). If you can debunk this rule, please do. I'm really interested in knowing if I've come up with something that, if not unique - is at least accurate.

Cheers,
Tim
 
Please don't make bet sizing decisions based on direct odds only. Hero has to bet enough to make the direct plus implied odds unprofitable.

Giving the villain almost exactly proper direct odds means he gets to value bet you every time he hits and/or bluff you when a scare card hits.

Hero also can face multiple types of draws. It isn't a sure thing that villain is drawing to 8 or 9 outs, could be more.

You bet what villain will call, but at least enough to cover the implied + direct risks. And that means Hero also bets the same way when bluffing or semi bluffing.
 
DrStrange said:
Hero has to bet enough to make the direct plus implied odds unprofitable.
Click to expand...

Exactly.

Even if, by some miracle, you can always know exactly how many outs your opponent has... betting just enough to deny them proper immediate calling odds simply sets you up to be eating alive over time - either by bluffing you off on scare cards, or by making you pay off when they hit.
 
DrStrange said:
Please don't make bet sizing decisions based on direct odds only. Hero has to bet enough to make the direct plus implied odds unprofitable.

Giving the villain almost exactly proper direct odds means he gets to value bet you every time he hits and/or bluff you when a scare card hits.

Hero also can face multiple types of draws. It isn't a sure thing that villain is drawing to 8 or 9 outs, could be more.

You bet what villain will call, but at least enough to cover the implied + direct risks. And that means Hero also bets the same way when bluffing or semi bluffing.
Click to expand...

Well. I certainly understand what you're saying but my example clearly stated implied odds weren't in play. I suppose you could start with this formula and then adjust for implied odds. I just think it gives me a good starting point. I appreciate the feedback though.
 
CHP TD said:
and at low limits there are players who cant fold. ugh...
Click to expand...

Definitely. However, if they don't fold when you're giving them poor odds - that's a win for you over time, right?
 
Poker is more than numbers.

For example. I think you already had your "excel wizardry" answer before you asked the question, and it was a look how smart I am type of question.

The best starting point in poker is understanding some people don't play like excel spread sheets. And some do.
 
OklahomaBill said:
Poker is more than numbers.

For example. I think you already had your "excel wizardry" answer before you asked the question, and it was a look how smart I am type of question.

The best starting point in poker is understanding some people don't play like excel spread sheets. And some do.
Click to expand...

Just once I'd love to open a thread where someone wasn't an ass. Thanks for your opinion though.
 
chkmte said:
Definitely. However, if they don't fold when you're giving them poor odds - that's a win for you over time, right?
Click to expand...

Poker is all about mistakes. As you note, placing a bet that gives the opponent the wrong odds to call... and then him calling is his mistake and you make money. Placing a bigger bet, up to his calling tolerance threshold... is a bigger mistake and you make more money. You want to bet as much as they are likely to call.

Additionally, against multiple opponents, placing such a small bet relative to the pot might price out villain 1, but as soon as he calls, villain 2 is getting a great price to call. In this situation, a pot sized bet is still giving villain 2 a 3:1 situation directly.

What is your justification for not considering implied odds?
 
OklahomaBill said:
Poker is more than numbers.

For example. I think you already had your "excel wizardry" answer before you asked the question, and it was a look how smart I am type of question.

The best starting point in poker is understanding some people don't play like excel spread sheets. And some do.
Click to expand...

Am I missing something? Why troll this thread?
 
chkmte said:
Just once I'd love to open a thread where someone wasn't an ass. Thanks for your opinion though.
Click to expand...

Well, was I right? Did you already have your answer in your pocket before you asked the question?

Not trying to troll the thread... And I apologize if it seems that way. strictly mathematically I think your calculations are accurate but incomplete by not considering all the factors that make up a hand.
 
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Shaggy said:
What is your justification for not considering implied odds?
Click to expand...

I'm not trying to justify "not" considering implied odds. There are so many factors in play when it comes to implied odds, it would have been impossible for me to ask the original question without leaving those out. In other words, I was simply trying to come up with a starting point for a bet. Obviously there is more to it than this.

What I've learned from this exercise is where to start. So, if I don't bet at least this much , I'm giving my opponent good odds. I hope that makes sense.
 
Da
chkmte said:
No. You were off the mark this time Uncle Bill - but troll as you please.
Click to expand...


Dang it... Your a profit / loss kinda guy... I figgured you knew your answer before you asked the question.

pocket aces against a open ended straight draw? I'm raising 25.80
 
chkmte said:
What I've learned from this exercise is where to start. So, if I don't bet at least this much , I'm giving my opponent good odds. I hope that makes sense.
Click to expand...

Yeah... but I would maintain that even by betting exactly "that much" you are still giving the opponent good odds. Think of it from the opponents perspective. He is drawing to an open end straight. 8 outs... 16%. 84:16 = 5.25:1. The pot has $50 in it. chkmte just bet 24% of the pot = $12. The opponent is thinking... I am getting the wrong odds now, but if I hit my hand, will chkmte call a turn bet of $1 in a pot of $74+$1... you bet he will... and he will call your $12.

Now let's look at it if you bet 2/3 the pot.
He is drawing to an open end straight. 8 outs... 16%. 84:16 = 5.25:1. The pot has $50 in it. chkmte just bet 2/3 of the pot = $33. The opponent is thinking... I need to win $33*5.25 = $173.25 in order to justify this call... Can I do that? I am getting the wrong odds now, but if I hit my hand, will chkmte call a turn bet of $90 in a pot of $116+$90 = $206... hmmm, I think he will... and he will call your $33. If you can find a fold when that card hits, good for you... if not, you just justified his call of $33.

The implied odds is absolutely crucial. The $33 bet is borderline... the $12 bet is way too small.
 
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Yeah, I get it @Shaggy . I understand where you're coming from and I appreciate your input. BTW, did you get those Cali Bells yet?
 
You all need to realize that both the question and the answer to that question were very specific. It asked you to ignore implied odds and I think also to ignore the river card (asked for a decision based on "immediate" odds). I think we all know that a lot more goes into decision making and bet sizing than just outs, especially in no limit. But the question asked was very limited and precise and that's what the answers should address.
 
Andrew Marks said:
You all need to realize that both the question and the answer to that question were very specific. It asked you to ignore implied odds and I think also to ignore the river card (asked for a decision based on "immediate" odds). I think we all know that a lot more goes into decision making and bet sizing than just outs, especially in no limit. But the question asked was very limited and precise and that's what the answers should address.
Click to expand...

exactly.
 

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