7

Flop = T♥ 7♥ 6♥

Player 1 = 4♠ 3♥

Player 2 = 5♥ 2♠

Whats the equity of Player 1 on this flop ?

Equity calculators give it 43.54% (18.5% for tie and 25.04% for win). No matter how hard i tried to break it down on simple probabilities and try to reach to the win equity of 25% i failed, closest i got was at 25.55% with rather spooky and most probably incorrect use of maths.

Here is my best try:

  • Player 1 wins IF 4 or 3 or 5 comes (not 4h) on turn or river AND avoid a heart on river AND avoid both 4 AND 3or8 on turn and river.

This breaks down in 3 probabilities: PA = 4 or 3 or 5 comes on turn or river (not 4h) = 8outs = 8/45 + 8/44 - 8*8/(45*44) = 32.73%

PB = not a heart on river = 1-8/44 = 0.8182

PA * PB = 26.777 % So this is presumably the probability that one of the 8 outs comes for Player 1 and avoids a heart on the river (now why to avoid a heart only on river i really dont know so thats why i think the calculation is not correct, but that was the only way to reach close to 25% if both turn and river were asked for i was getting a Probability of win close to 21% only)

PC = turn or river 4 AND 3or8 ( not 8h) That was another problematic probability that i was not sure how to handle. The way i decided to handle it is the following:

  • 8outs on turn and IF 8 or 3 then 2outs on river, or-else if 4 on turn 6 outs on river

PC =(8/45) X [(6/8)X(2/44)+(2/8)X(6/44)] = 0.01212

Hence Pwin of player 1 = PA*PB-PC = 26.777-1.212 = 25.56%

Is this correct ? Most probably not. What am i missing ? What would be an easy way to calculate correctly this?

8

You should break it into disjoint (non-overlapping) cases, and find the probability that you win with each case, and then add them up:

Case 1. Heart on turn

Occurs 8/45 times and your probability of winning is 0

Case 2. 8 (not heart) on turn

Occurs 3/45 times and your probability of winning is 6/44 (three 3's and three 5's)

Case 3. 5 on turn

Occurs 3/45 times and your probability of winning is 36/44 (Any non heart)

Case 4. 4c or 4d on turn

Occurs 2/45 times and your probability of winning is 30/44 (No heart, 8, or 3)

Case 5. 3 on turn

Occurs 3/45 times and your probability of winning is 34/44 (no heart or 4c or 4d)

Case 6. Not a heart, 8, 5, 4, or 3

Occurs 26/45 times and your probability of winning is 8/44 (hit one of your eight outs on the river)

Overall probability of winning:

 8/45 *  0    + 
 3/45 *  6/44 +
 3/45 * 36/44 +
 2/45 * 30/44 +
 3/45 * 34/44 +
26/45 *  8/44 = 0.25050505

The left column should add up to 1, since each case is disjoint and we've covered every case. The overall probability is 25.05%. I'm not sure why you're calculator said 25.04%, but I've verified that 25.05% is the correct win%. Perhaps you were using a monte carlo method or it simply had a rounding error.

The way to come up with all these cases on your own, is to start small, say by only considering two cases.

Case 1. 5 on turn

3/45 * 36/44

Case 2. Anything else

42/45 * X 

X is hard to calculate, so we should divide this case into more cases, maybe we'll add a 3 on turn case next. Then repeat, adding cases, until each case has a simple to compute probability of winning. You could do this whole procedure in a similar way to calculate your probability of a tie, and then use win% + tie%/2 (assuming only two players) to calculate your actual equity.

  • I think the small difference between your calculation and the calculator is that you are not considering the case that two big hearts are dealt in turn and river which splits the pot. In case 1 you say any heart on turn means you loose. That is not the case any two cards from 8 9 J Q K and A of hearts in turn and river splits the pot. Haven't done the math but that probably can explain 0.01% difference. – Mohsen Nosratinia Jan 22 '16 at 12:44
  • @MohsenNosratinia I'm only calculating the chance of winning the whole pot and not the chance of a tie in this answer. The win% is still 0 when there is a heart on the turn. A tie is still possible, but a complete win isn't possible any more. Notice that in the OP's question the total equity including ties is 43.54%, but they were only trying to calculate the portion of that which comes from winning. – Paulpro Jan 22 '16 at 15:49

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