OK, so either I'm missing something, or this odds generator is off. My understanding of the 4x2 rule is that your equity (after the river) can roughly be calculated by multiplying your outs by 4 after the flop and 2 after the turn.

So, in the hand in the image attached, I calculated my equity as being ~60%

(three 5's, two 7's, ten spades=15 outs; 15*4=60)

This odds calculator has me at 51%

who's off here?

http://imgur.com/a/21lyj

You can hit some of your outs and still lose those times that your opponent also hits their outs is my response having not done the math counting all possible board runouts.

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To clarify, your equity is equal to your chance of improving your hand by hitting one or more of your outs if and only if you are guaranteed to win the hand when you hit one or more of your outs. When your opponent has redraws, your equity must be lower than the chance of improving to a better hand.

Having not done the math, I'm guessing that accounting for board runouts in which, for instance, you improve to two pair or trips but your opponent improves to a better two pair or trips, or you improve to a flush but your opponent hits running cards to a full house, will put your equity closer to what the calculator says it is.

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So first, you miscounted your outs. You have 9 spades not 10, a total of 14 outs

Also the rule of 2 and 4 is an estimate, and the rule of 4 gets less accurate the more outs you have.

So let's say you know your hand and the board, that's 5 cards with 47 remaining. You have X outs. Your chance of getting your hand in one card is X/47. X/47 is kinda close to X/50 and X/50 = rule of 2. So, close enough for most things.

You chance of getting it within 2 cards is equivalent to 100% minus the chance of missing both times. That is
1 - (47-X)/47 * (46-X)/46

In your particular case, this would be
EV = 1- (47-14)/47 * (46-14)/46 = 51.15%

You can see why the rule of 4 gets worse the more outs you have, hopefully. Let's actually consider a simpler situation where your chance of winning stays the same on each card (it changes very slightly on each street in holdem when you miss because you have the same # of outs, but fewer cards to choose from)

Let's instead say your chance of winning in one card is W. The chance of winning when getting 2 cards would be close to
EV = 1 - (1-W)*(1-W)
So if W=10% then we have
EV = 1- .9*.9 = 1-.81 = .19. This isn't too far from the estimate of 2*W which would be .2

But what if W is, say, 60%? Clearly our chance of winning can't be 120%. Actually, it's
EV = 1 - .4*.4 = 1 - .16 = ..84

So using double your one-street equity for 2 streets is a good estimate only if your number of outs is small. At W=20% we have
1 - .8*.8 = .36 vs .4 which is, imo, about as far from reality as we'd want to get. So anything up to 10 outs, give or take. Above that I consider the rule of 4 pretty useless.

But also, at more outs than 10, we have a really good chance of winning and our exact equity is not as important, since pot odds will usually justify a call.

Thank you for the thorough explanation; I get it now.

You can also use a slightly more accurate version of the 4x2 rule which says to subtract the number of outs you have above 8 from the final percentage.

E.g. When you have 14 outs:

14 x 4 = 56 - (14 - 8) = 50%