What's wrong with my equity calculation?

Question

When answering this question, I went ahead and tried to do a manual equity calculation for the scenario described (7♥4♠ vs A♠8♣ on a board of 6♥2♥8♥), but am getting a different result than I see using any online equity calculators, so wanted to see if anyone can spot where I'm going wrong:

To win, the 7♥4♠ has the following options:

Make a flush (without villain making a full house)

This would require at least one of the turn or river to be a heart, plus discount the scenario where one of the turn/river are A♥ and the other is an 8 or another A:

[(Chance of heart on turn) + (Chance of heart on river if no heart on turn*)] - (Chance of Ah and A/8 on turn/river)
= (9/45) + [(36/45) * (9/44)] - {[(1/45) * (4/44)] + [(4/45) * (1/44)]}
= 0.2 + [0.8 * 0.2045] - {[0.02222 * 0.09091] + [0.08888 * 0.02273]}
= 0.2 + 0.1636 - {0.002020 + 0.002020}
= 0.2 + 0.1636 - 0.004040
= 0.3596 (35.96%)

*we discount instances where a heart already came on the turn because those are included in the (Chance of heart on turn).

Make a straight

This can be done either by hitting a 5:

(Chance of non heart 5 on turn) + (Chance of non heart 5 on river where 5 or a heart didn't come on turn)
= (3/45) + [(33/45) * (3/44)]
= 0.06666 + [0.7333 * 0.06818]
= 0.06666 + 0.05000
= 0.1167 (11.67%)

Or runner runner 9T/T9:

(Chance of non heart 9/T on turn) * (Chance of non heart 9/T on river not pairing turn)
= (3/45) * (3/44) * 2 <- Doesn't matter if it goes 9T or T9
= 0.06666 * 0.06818 * 2
= 0.009090 (0.91%)

Hit runner runner trips

This can be done with running 44 or 77:

[(Chance of 7 on turn) * (Chance of 7 on river)] + [(Chance of non heart 4 on turn) * (Chance of non heart 4 on river)]
= [(3/45) * (2/44)] + [(2/45) * (1/44)]
= [0.06667 * 0.04545] + [0.04444 * 0.02273]
= 0.003030 + 0.001010
= 0.004040 (0.40%)

Hit runner runner 2 pair

[(Chance of 7 on turn) * (Chance of non heart 4 on river)] + [(Chance of non heart 4 on turn) * (Chance of 7 on river)]
= [(3/45) * (2/44)] + [(2/45) * (3/44)]
= [0.06666 * 0.04545] + [0.04444 * 0.06818]
= 0.003030 + 0.003299
= 0.006329 (0.63%)

So summing all of these probabilities, we get:

35.96% + 11.67% + 0.91% + 0.40% + 0.63% 
= 49.57%

The result I get from online tools is 48.18%, so I must be missing some redraws which the A8 has (I don't see any except hitting a boat when hero makes a flush with the Ah), or there's a fault in my calculations. I am rounding to 4sf, but I can't see this accounting for a difference of almost 1.4% on this calc...

Answer 1

At first, I thought, OK, lets quickly answer this, returning the favor. But it turned out to be a rather tricky one! One online calculator gives 48.18%, the other 48.28%, off by 0.1%, already very strange. My first result was 49.09%, off by exactly the runner-runner 0.909%. You also confused me with your cases and some calculations.

The tricky part is - of course - the Ace of Hearts. The solution is best found by considering different turn cards separately.

So here we go:

First we consider incontestable outs, i.e. the ones without redraws. Any heart other than the Ah will do, plus the 3 off-suit fives. Hitting one of these 11 outs on the turn obviously gives equity of:

11/45 = +0.24444

Now for the Ah on the turn, where we have to fade a 4-out redraw (A,A,8,8). So we have:

(1/45) * (44-4)/44 = +0.02020

Now for the remaining 33 turns! (No direct out and no Ah) We have to do one more split, to crack the problem!

First, the 29 safe turns, where we can win on the river with any heart or five. Thus, no A or 8 on the turn. There are 33-4 = 29 such turns, so we win on 29 turns by hitting one of our 12 outs (including Ah) on the river:

(29/45) * 12/44 = +0.17575

Second, the 4 unsafe turns (A,A,8,8, no Ah), where we only win by hitting one of the 11 incontestable outs (without the Ah). So we win on 4 turns hitting one of 11 outs:

(4/45) * 11/44 = +0.02222

Now we have considered all possible turns.

What remains are runner-runner wins, which are - luckily - independent of the above problems, not influenced by the Ah, so we can just add their equity. We just have to make sure that we do not count in any hearts, as they are already accounted for above.

Add runner-runner T9 (factor 2 used here, as you correctly note, for order does not matter). We have 3 non-heart tens and nines, so

3/45 * 3/44 * 2 = +0.00909

Finally add runner-runner trips or two-pairs (no need to split these cases). We have to hit one of 5 cards (3 sevens, 2 fours, no 4h) on the turn and another one (4 remaining) on the river !

5/45 * 4/44 = +0.01010

Summing up, we have

  0.24444 (11 incontestable outs on T)
+ 0.02020 (Ah on T)
+ 0.17575 (safe T, hit 12 outs on R)
+ 0.02222 (unsafe T, hit 11 outs on R)
+ 0.00909 (runner-runner T9)
+ 0.01010 (runner-runner 77,74,44)
= 0.4818

In line with the online result.

Moral: Thou shalt not calculate equities manually!

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About 'what went wrong in your calculation':

The part 'Make a flush' correctly includes the cases where we lose to the Ah on the turn and to the redraw. The second case of (A,8) first is also taken into account.

The part about the straight is also clearly ironclad.

What you have been missing is a subtlety often missed in probability. Things have to be independent in order to add them up!

So where is the dependence, you rightfully ask, I don't see any!

Here it is: If you have already treated straights separately, you have to take this into account when doing the flush calculation. As strange as it might seem at first. The spot where it is of importance is the part where you 'missed' on the turn and need to hit on river. You take it as

[(36/45) * (9/44)]

and add this to your overall equity. In fact, you are only allowed to use

[(33/45) * (9/44)]

because the 3 straight out are already accounted for in your 'straight' part!!

My approach did not suffer from this problem because I treated flush and straight outs the same.

The discrepancy is thus

(3/45)*(9/44) = 0.013636

together with the minor 0.0003 you overshot (by a typo) in the runner-runner two pair, this will make the desired result!

Answer 2

I'm not sure how much it affects your equity calculation, but you are missing where the villain hits Aces full.