calculating EV, 3-way all in preflop

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Rising Star
Mar 8, 2021
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All in preflop against two:
me: QsQd, 114.5 BB, 48.52% equity
against: AsTh, 60 BB, 24.63% equity
and also: KcTc, 49 BB, 25.3% equity

I want to factor into the EV the times when I make 2nd best hand, losing 49 BB and winning 11 BB for a net loss of 38 BB. As I have yet to acquire a poker book that is math-heavy enough to go into these finer details, here I am trying to figure it out myself:

Suppose it were only AsTh and QsQd getting all-in preflop, while the Kc and Tc are dead cards. Then Queens beat the Ace-high 70.63% of the time. This leads me to think that in the 3-way situation, 70.63% of that 25.3% when the King-high wins, I beat the Ace-high and make 2nd best, which would be 17.9%.

If that happens to be right, I expect to win on average:
0.4852 * 109 BB + 0.179 * -38 BB + 0.2463 * -60 BB = 31.3 BB

I've taken my time, writing my first post. Is my thinking flawed or not?
Phoenix Wright

Phoenix Wright

Feb 18, 2020
Total posts
Your math is a lot closer than I'd probably calculate out myself :D

According to the free poker calculator download Equilab, the equity here is actually:

QdQs: 48.62%
AsTh: 25.35%
KcTc: 26.03%

This only applies to this exact situation with these exact holdings though. In the "real poker world", we do not know the opponent exact two cards. We must then calculate the range of likely hands we suspect them to hold and see how our exact cards compare to the range of hands they may plausibly have.

In this example with us holding pocket Queens, we do not know if we are up against KK or AA. Players would probably play these from any position at the table. Queens are still great though, the opponent(s) probably have something worse right now (and they did in this example as QQ is favored).

Now we get more into the realm of strategy asking about what to do, but the more players in the pot, then more less equity your hand has. In this example, one opponent held an Ace and one opponent held a King, so now any overcard gets ahead of our QQ. This is why "isolation" (raising to get less players to contend with your hand at showdown) can be a powerful tool in a poker players' arsenal. You said you were All-in so this doesn't directly apply here, but if we somehow even got just one of the other opponents to fold, now we have a much better chance of winning.

Let us assume that somehow we got here, but with the KcTc player to fold.

Now the equity would be:

QdQs: 70.89%
AsTh: 29.11%