Equity calculation

Question

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?

Answer 1

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.