Expected value and bluff betting. Math that drives me crazy.

    • nedostoevsky
      Joined: 26.11.2013 Posts: 3
      I really can't understand one thing...

      There are two players at the flop. Let's imagine that there are $100 in the bank and we make a bet of $50. We are calculating fold equity of bluffing bet(there are two options in this model - opponent folds - then we win, or opponents raise - then we fold and lose)

      fold EV = bank*(probability that opponent's fold) - bet*(probability opponent raise)

      Is this right?

      If so, here is a thing confusing me.

      Let's imagine the probability that opponent will fold is 100%. This means that every game, on average, our profite = $100.

      But this is nonsense! The very maximum of our profit can be only $50(the villains part of the bank).

      The thing that confusing me is the rule, according to which we should bet if ev(bet) is positive.

      Lets imagine - there are $100 in the bank. Our bet is still $50. Now the probability that opponent will fold is 35%; raise is 65%.

      ev= 100*35%-50*65% = $2,5 - Looks good, so we should bet.

      But let's now look from the profit inflow\outflow position:

      The maximum net profit we can make is still $50 - the villians part of the bank;
      The maximum we can lose = our part of the bank + our bet $50+$50=$100

      Let's calculate net inflow now with the same 35% of fold and 65% of raise.

      net profit = 35%*50+65%*100 = -$47,5 - looks like a disaster now!

      So, there is a clear contradiction and I don't know how to solve it myself.
  • 6 replies
    • TinoLaan
      Joined: 12.10.2011 Posts: 6,411
      Hey nedostoevsky,

      First of all, welcome to PokerStrategy.com!

      You are in a sense correct. I'm just going to list a few assumptions here for your case.

      1) The pot is $100 and we bet $50.
      2) Villain never calls, but only raises or folds.
      3) We always fold if villain raise.

      These are the assumptions you're working with for this example, right? Now let's say villain folds 80% of the time and raises 20% of the time. So 80% of the time we win the $100 that's in the pot, and 20% of the time we lose out bet of $50. So the EV is

      EV = .8 * $100 + .2 * -$50 = $80 - $10 = $70

      As a formula:

      EV = P(villain folds) * pot + P(villain raises) * -bet
      EV = P(villain folds) * pot - P(villain raises) * bet

      where P simply means "probability".

      So if villain folds 50% of the time (and therefore raises 50% of the time), your EV will be:

      EV = .5 * $100 + .5 * -$50 = $50 - $25 = $25

      But this is nonsense! The very maximum of our profit can be only $50(the villains part of the bank).
      This is where you are wrong. Any money you put in the pot on earlier streets is no longer yours. In fact, it belongs to nobody. Therefore it will be up for grabs again for you.

      Say you start with a $1000 stack and put in $50 pre flop. You now have $950 left. If villain now folds to your bet on the flop, you will have a $1050 stack. Yes, that's a total profit of $50 in the end, but you only had a $950 stack on the flop. So in a sense, you made a "profit" of $100.

      So for your second example, where villain folds only 35% of the time (and thus raises 65% of the time), your EV is (like you already calculated)

      EV = .35 * $100 + .65 * -$50 = $35 - $32.50 = $2.50

      So you still make a profit.

      Hopefully that makes sense! Just drop us another message here if anything is still unclear :)

      Kind regards,
    • VorpalF2F
      Super Moderator
      Super Moderator
      Joined: 02.09.2010 Posts: 8,828
      Hi, nedostoevsky,
      There is an article in the strategy section that deals with this, but you'll need higher status.

      Fortunately, higher status is fairly easy.
      Pass the quiz, then get some free poker money.
      Once that's done sign up with an affiliated Poker Room and you'll be bronze level.

      Best of luck!
    • nedostoevsky
      Joined: 26.11.2013 Posts: 3
      Thank you for your answer, TinoLaan, I think I got the idea :)

      I have one more question realated to the topic:

      how does a formula that answer the following question looks like?

      What is the probability that opponent will fold at the flop must be that making our expected profit value => 0?

      Like there is $100 in the pot and we are thinking about betting 2/3 of the pot.

      So what is the minimum p. of fold should be there that make our EV=>0?

      What's the formula? (going to put it in my excel working list)

      (if opponent make a raise we still fold; P(raise) - is a constant; P(call)=1-P(fold)

      and can we take into the model 8 outs for a good straight we can get at the turn?)
    • TinoLaan
      Joined: 12.10.2011 Posts: 6,411
      Like there is $100 in the pot and we are thinking about betting 2/3 of the pot.

      So what is the minimum p. of fold should be there that make our EV=>0?
      That's relatively simple. In order to automatically show a profit with a bet (i.e. the bet is +EV regardless of the cards you have), your opponent has to fold an x percentage of the time. That depends completely on your bet size.

      You can find out how often a bet needs to work to show an auto-profit using this formula:

      bet / (bet + pot)

      So if we bet $25 into a $100 pot, we need the bet to work $25 / $125 = 20% of the time. If we instead bet $50, our bet needs to work 33% of the time to show a profit with any two cards. If you bet $50 and your opponent folds 40% of the time, you show an immediate profit regardless of the cards you are doing it with. So even if he does call sometimes, which will of course happen, you can still cbet your entire range and be +EV.

      It isn't quite that easy on the flop and turn though, because you still have multiple streets to go so there are many other factors to take into account. However, you can probably apply it in a vacuum to both the flop and the turn. It certainly applies on the river.

      Also worth noting that even though you might show an auto-profit with a bet (i.e. the bet is +EV), betting might not be max EV with your entire range. For example, it might be better to check/call or check/raise some hands, instead of betting them yourself.

      I know I'm straying away from the formula you were asking for a bit, but I thought this might be useful information for you to have :)

      Either way, an EV formula for betting on one street looks like this. Let's assume for now that this is the river, and like you said we fold if villain raises.

      EV(betting) = P(villain folds) * pot + P(villain calls) * [(equity * pot + 2 * bet) - bet] - P(villain raises) * bet

      So, let's now assume villain folds 40% of the time, raises 10% of the time, and calls 50% of the time. If villain calls, we have 70% equity (so we win 70% of the time and lose 30% of the time). The pot is $100 and we bet $50. Our EV for betting on the river will then be:

      EV(betting) = .4 * $100 + .5 * [.7 * ($100 + $50 + $50) - $50] - .1 * $50
      = $40 + $45 - $4 = $80

      Now let's say we bet $100, so we bet full pot. Now villain will fold much more often, so 70% of the time. He will still raise 10% of the time, and when villain calls (20% of the time) we expect to be behind fairly often, we only have 30% equity this time when he calls. Now our EV looks like this:

      EV(betting) = .7 * $100 + .2 * (.3 * $300 - $100) - .1 * $100
      = $70 + $0 - $10 = $60

      Hopefully that answers your question!

      and can we take into the model 8 outs for a good straight we can get at the turn?)
      I'm not quite sure what you mean here. If you mean the equity a straight draw gives you, that's already determined in the above formula with the equity we have when villain calls.

      Hopefully that made sense, and sorry for the long post :f_biggrin:
    • nedostoevsky
      Joined: 26.11.2013 Posts: 3
      Thanks for your detailed answer, TinoLaan!

      I am sorry I took so much of your time!

      The formula - bet / (pot +bet) is very simple to work with :)

      I'd love you to tell me if my logic is correct.

      Let's imagine that our hand is 78 and there is 69QK on the turn.
      Let's also imagine that we know that our opponent hand is Q2.

      There are $200 in the bank and we are thinking of making 2/3 pot bet.

      According to the formula the opponent should fold for 133/(133+200) = 40% on the turn. But, if the opponent call, we still have a chance to win with a straight.

      value added by straight came in 60% cases when our bet called (200+133)*17,4%*60% = 34,78

      If we have a 0 potential hand (that mean that our bet is a bluff)

      the villians-fold-%-required = bet / (bet + pot) = 40%

      But we have a hand with some potential, so the formula can be adjusted (in order to lower the %-of-opponet-fold-required, as we have much chanses to win compared to bluff betting)

      Thus, if %-of-opponent-fold is not 40% but 37%, we still have a positive ev

      37%*200-63*133 = -9,8 [this is our EV with excluded hand potential]

      But a straigh add value:

      (1-37%)*(200+133)*17,4%= 36,5 [this our pure hand potential]

      So adding them up we notice that the number is positive, and less then 40% of fold is enough to make a bet

      So my question is: is there a way to add the potential hand value in formula [ bet / (bet + pot)] in order to get the lowest %-of-fold-requierd to be >=0
    • TinoLaan
      Joined: 12.10.2011 Posts: 6,411
      No worries, glad I can help! :)

      And holy crap this post is even longer than my previous one :O

      That is pretty much exactly correct. Let's just make a very simplistic example here to show this. We'll assume the board and the hands you mentioned, so 87 and Q2 on a KQ96 board, assuming no flush draws. Now let's also say that our bet is always a shove here, so there is going to be no further action on the river. The pot is $300 and we will shove $200 (so a 2/3 pot bet like you mentioned). We'll also assume that villain always calls our shove.

      We are going to win 18% of the time. So a formula to calculate the EV of betting here would be:

      EV(betting) = .18 * $700 - $200 = -$74

      By the way, if you didn't know this yet, you can find out the equity your hand has against a specific hand (or a range of hands) on a specific board using for example the Equilab. This is a very useful tool to have, you can find more info here: Equilab

      So, like you said, if you want to know how often a bet needs to work in order to be profitable, you use the formula bet / (bet + pot).

      Regarding what you asked here:

      Are there a way to upgrade formula with the odds we have?
      That's definitely possible, but that's just an incredibly large EV calculation. Even the one I gave in my previous post is a good example.

      Also, above I gave a very simplistic example, where we know the exact hand villain hands. Realistically, this is almost never going to happen. Instead, you are always playing against a range of hands, that's how you should always try to think. The Equilab is a great way to practice that.

      So let's take the above example again, and do a more detailed EV calculation.

      We have 8:h7 on the BU and make a standard raise of 2bb. BB calls, who has only a small stack of 25bb. Let's for the sake of simplicity say we have played thousands of hands with him already, and know he calls with the following range:


      Then the flop comes Kh9:6c: and there's now 4.5bb in the pot.

      He checks, we fire a cbet of 3bb and he raises to 9bb. Now let's try to figure out what the EV of shoving here is.

      Let's assume that our opponent only raises with flush draws, straight draws that are not gutshots, two pairs, sets, and top pair with a J or Q kicker. That means our opponent's range is this:


      That is a total of 54 hand combinations.

      Let's also assume that he folds his top pairs, 87, and 65s when we shove. When we put these ranges in Equilab, we see that we have roughly 49% equity against his range with 87 of hearts.

      So when he calls our raise, we are up against this range:


      That's a total of 26 hand combinations, and we have 41% equity against this range. So an EV calculation would look like this. He folds the other 28 hand combos, so he folds 28 / 54 = ~52% of the time.

      After his raise on the flop, there is 4.5 + 12 = 16.5bb in the pot (let's ignore rake for now). So if villain folds, we win 16.5bb. We have to put in another 20bb. Let's sum up what we know:

      - Villain folds 52% of the time, which means that 52% of the time we win 16.5bb.
      - Villain calls 48% of the time, and we will win a 50.5bb pot 41% of the time, and lose our 20bb the other times.

      So the calculation would look like this:

      EV(shoving) = P(villain folds) * pot + P(villain calls) * [(P(we win) * final pot + P(we lose) * our investment)]

      So to put the numbers in there:

      EV(shoving) = .52 * 16.5 + .48 * [(.41 * 50.5) + (.59 * -20)]
      = 8.58 + .48 * (20.705 - 11.8) = 8.58 + 4.2744 = 12.8544

      So in the end, we expect to make almost 13bb by shoving here.

      So this way you incorporate all the equity your hand has, compared to what your opponent has.

      And that's still only a very simple example. Hopefully this illustrates the point though :)

      Finally, you can learn a lot of these types of things in our strategy articles and videos. Of course, you are still a Basic member so you do not have access to most of our content yet. You can do this by either getting our free poker money (which is always a great deal), or by signing up with one of our partner rooms and making your own deposit. Either way, playing tracked on a poker room will unlock more content for you, which should certainly be useful for you :)

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      I'll really stop typing now :f_biggrin: