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...