Can someone help with this problem from "Poker Math that Matters"? What would the all-in equity be for the following:

Hero: J♠A♠

Villain Range: QQ, KK, 55, 44

Board: 5♦4♠T♠

I'm just starting to apply math to poker and I can't wrap my head around this.

I can figure out the equity if Villain has QQ or KK. Around 44% ((8outs if Villain has a spade + 3 Aces) x 4).

What I can't figure out is the equity vs a set. Don't know, just can't count my outs. A calculator will give me around 25%, meaning 6ish outs(?). How did it come up with this?

The way I'm trying to figure out my equity overall is:
1.Determine the equity against each hand in the range.
2.Multiply the equities by the number of combinations for that hand.
3.Add the results from step two for each hand in the range.
4.Divide the results from step three by the total combinations in the hand range.

So I'm stuck on step one. How do I count my outs if the villain has 55 or 44 on that board?

Welcome to the Forums.

If villain has a set, you not only need to hit your flush, you also need villain to not make a full house or quads. So yes, you still have roughly 8 outs, twice, to hit your flush (so roughly 32% equity) but his redraw potential hurts your equity. He has 7 outs on the turn and, assuming he doesn't improve on the turn, 10 on the river to improve to quads or a full house.

So in this case, it's more complicated than just counting outs on the flop and multiplying by four.

ProPokerTools Odds Oracle Results (2.27 Professional)
Holdem, Generic syntax
Board - 5d4sTs
PLAYER_1 AsJs
PLAYER_2 5s5
1980 trials (exhaustive)


All-in Equity

	Equity %	Wins Hi %	Ties Hi %	Wins Hi Count	Ties Hi Count
AsJs	26.4646%	26.4646%	0.0000%	524	0
5s5	73.5354%	73.5354%	0.0000%	1456	0

EDIT: Just realized, you also have a couple of backdoor straight draws.
Even if you don't hit your flush, runner-runner KQ or 32 would both give you a straight