Part five.

Previously:
Intro and Probability
Combinatorics
Betting Odds
Pot Odds

index of all the parts of this series

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EQUITY - the rule of '2 and 4'

Suppose you get to the turn, holding the nut flush draw - A5 on a K47Q board.
You’re heads-up against a nitty Villain who never puts money in the pot without a strong hand. He bet the flop and has now shoved all in on the turn.
That makes the pot $80 and it will cost you $30 to call.
Should you call or fold?
We’re confident, given our read on Villain, that’s he’s not bluffing. He has at least two pair in this spot meaning we need to hit our flush in order to win the hand.

Let’s think about the odds on offer.
With no future betting possible, calling would risk $30 to win $80, so we’re getting odds of 80:30 or 8:3.

How many cards make our hand?
There are 46 unseen cards.
Of those, any of the 9 remaining hearts will make our flush but the Q, or the 7 would pair the board and very possibly give Villain a full house or quads so we decide not to count those amongst our ‘outs’.
That leaves 7 cards that would give us the winning hand, and 39 that won’t.
So the odds against us winning the hand are estimated as 39:7 or 5.57:1

The pot odds we’re being offered are 8:3 or 2.67:1
In other words, the pot odds aren’t big enough to justify a call.
We should fold.

Suppose the pot was $280 after Villain shoved, and it still cost us $30 to call.
In this case, the pot would be offering us odds of 280:30 or 9.33:1, much larger than the odds against us making our hand, so we’d have an easy call.

So one way to make this kind of decision is to compare the pot odds we’re being offered with the odds against making our hand. If the pot odds are bigger than the odds against making our hand, we should call. Otherwise we should fold.

The other way of thinking about these decisions is to consider our equity.
Equity is just a fancy word for the probability that we’ll win the hand.
In this case, we're assuming that just 7 cards out of 46 unseen cards give us the winning hand.
So our equity is 7/46, or roughly 15%

The odds on offer (in the original case, where the pot was $80 and we could call for $30) were 8:3 which means we need to be good 3/11 of the time, or (approximately) 27%

So again, we should fold, since our chance of winning (our equity) is much smaller than it needs to be for calling to be profitable.

For the sake of completeness, note that when the pot is offering us odds of 280:30, we need to be good just 30/310 of the time or (approximately) 10%, which is much less than our equity of 15%, so we have an easy call.

The following table shows how the probability of making our hand on the river changes with the number of outs we have to make the best hand:



This gives rise to the ‘rule of 2’ which says that with one card to come, we can estimate our % equity by simply multiplying our number of outs by 2. (In other words, the numbers in that last column and 'close enough' to those calculated in the middle column.)


What if Villain shoved all in the flop, rather than the turn?
It’s more complicated this time because there are two cards to come, not one. To work out the probability that we make our hand, it’s probably easier to work out the probability that we don’t make our hand, and subtract that from 1.

There are now 47 unseen cards.
Suppose we believe that we have 7 outs. (I haven’t given an example hand here, but it’s not important. Let’s just say that we’ve given ourselves 7 outs, for the sake of argument.)
The probability that we don’t hit on the turn is 40/47.
If we don’t hit on the turn, there are then 46 unseen cards remaining, 7 of which would be good cards for us on the river. So the probability that we don’t hit on the river would be 39/46.

The probability that we miss on the turn and then miss again on the river is then the product of these two probabilities. ie. (40/47) x (39/46)

So the probability that we do hit on either the turn or the river
= 1 - the probability that we don’t hit on the turn or river
= 1 - (40/47) x (39/46)
which is roughly 0.28 or 28%

The following table shows how the probability of hitting on the turn or the river changes with the number of ‘outs’ we have to make the best hand:



This gives rise to the ‘rule of 4’ which says that with two cards to come, we can estimate our % equity by simply multiplying our number of outs by 4.

As always we have to remind ourselves that we’re making some pretty big simplifying assumptions in all of this. For a start, we’re assuming that we’re behind, and that we need to improve in order to win the hand. We’re assuming that Villain isn’t bluffing (I’m in no way trying to advise you guys on strategy, but clearly, the more bluffs we put in Villain’s range, the more rebluffing has to be part of our response, rather than simply calling and giving up if we don’t hit). We’re also assuming that we can calculate accurately our number of outs but it should be obvious that we’ll never be able to do that exactly because we never know exactly what hand Villain is holding. In other words, we’re always working with estimates. (But that’s fine. If we can make better estimates than our opponents, and/or use the results of our deliberations better than they do, we’ll make $ in the long run.)

So, we now have our ‘rule of 2 and 4’ which tells us how to estimate our equity when we need to improve, with 1 or 2 cards to come.

In all of the above we assumed that Villain was all in, so there was no further betting. Our decision in these spots was relatively straightforward. But what if Villain bet, and there was significant money still behind? In other words, what if there’s a very real chance that if we call, there could be further betting on later streets?

The most important thing to realise is that with the possibility of further betting, you should take things one street at a time. Consider the pot odds, and how they compare to your chance of hitting your hand (found by multiplying your outs by 2) on the next street. If you can call profitably, you should do so. If not, you should fold. (But see below for a reference to implied odds.)

If you call on the flop, because it was profitable to do so, but miss, and now face another bet on the turn, you can then rethink, and estimate your chance of hitting on the river, again by multiplying your number of outs by 2. In other words, take it one street at a time.

Facing a bet on the flop with, say, 8 outs, and thinking hey, I’m going to make my hand 8x4 = 32% of the time, so I can call could be a big mistake since that would be assuming that you see the next two cards for the price of that single call on the flop, and with further betting to come after you miss on the turn, that isn’t true.

To summarise, if Villain is all in on the flop, there are two cards to come, so you multiply your outs by 4 to get a good estimate of your equity. If he's all in on the turn, you multiply your outs by 2 to get a good estimate of your equity. Compare your equity to the odds on offer to decide if you should call or fold. In any other spot, where there’s the possibility of future betting, take it one street at a time, multiplying your outs by two to estimate the chance of hitting on the next card.

It’s important to remember that we still haven’t talked about implied odds. While future betting might make life difficult for us after we miss on the turn, future betting is great if we hit. So although in some spots pot odds and the rule of 2 might indicate a fold, we can sometimes still call profitably because of implied odds. We’ll look at implied odds next.

Your articles are really well written!

I remember reading that the rule of 2 and 4 becomes inaccurate when having more outs then 10. I can´t find the article anymore.

For the OESFD you have 15 outs that would mean 60%. Pokerstove gives you 54%. As a correction factor you should deduct the number of outs higher than 10 from your % results. That means 60%-5%= 55%.

Surely there is a more elegant way to calculate this (assuming this is and my pokerstoving is correct) but math and especially statistic was never my favorite subject.

Thanks for the feedback.

Yes, you make a very good point - for large numbers of outs, the rule of 2 and 4 will overestimate your equity, and one correction is to do what you suggest.

Having said that, of course, when our equity is around the 50% mark, we're going to make fewer, and smaller, in calling, compared to the hash we can make of things when we have, say, just 25%.

I've extended the original table to allow for more outs, and to show the effect of your correction:

Since odds, equity, and probability all come to down to assumptions about the other person hands, IE he doesn't bluff and has a strong hand, then it would seem to me you could also update your odds because of this.

For instance, going with your first example, we use 46 as the amount of cards left, 52 cards in the deck - 2 in our hand - 4 on the table = 46. We include our opponents cards as part of the cards able to come because we don't know what he has. I would argue that since our villain is playing strong hands, we may assume he has K, effectively pushing our 46 cards to draw from down to 45, thereby giving us a slight better, and if we are pretty sure about his second card then then our total cards would go down to 44.

This could effectively change your probability by near 1%. Not a huge difference, but a difference nonetheless. I just haven't seen anyone do calculations of probability to hit including the villains hand, but all equity calculations are based upon his hand.

Just curious about your thoughts about if you think we should update our calcs.

.. should have read 'fewer and smaller mistakes, in calling'

@Jefferey15: I agree - if we can put villain on precisely Kx (and in the context of my example that's not unreasonable) then we could, indeed, adjust our calculations to allow for the fact that there's one less non-heart in the deck. Good point.

Like all models, the rule of '2 and 4' is only as good as the assumptions we make. I'm trying to present it here as just a useful starting point in trying to get a feel for equities when needing to improve.

Oh I think the rule of 2 and 4 is good, and I agree it is good to use, especially in live games. I was just curious as to your additional thoughts about changing the calcs.

Solid post again. I hope all 5 parts of this series will be added to the FAQ.

Awesome post

Very good post and nice series. Looking forward to its continuation - I'd really like to see how you'll introduce implied odds.

Solomons rule, if you have more than 8 outs, subtract the number over 8 from your 4x answer. So 15 outs, 4x = 60 then take off 7 = 53

Nice.

very very nice article

Awesome thread.
Sooo... subract outs over 9 almost hits the percentage on the head, no?

great post

I wish people stopped talking about "the rule of 4". There is only one rule, it's the rule of 2, and it consists of multiplying your outs by 2 FOR EACH CARD TO COME. You're on the turn? 1 card left to come, outs * 1 card * 2. You're on the flop? 2 cards left to come, outs * 2 cards * 2. It can't be any simpler.

Moreover, if you want to know the chances of making a pair on the flop with your non-paired hand? If you learned "the rules of 2 and 4" you're completely on your own. If you actually understand the true rule of 2, you know you have 6 outs * 3 cards to come * 2 which is 36%. Real odds are about 32%.

Throughout DiamondDog's post I don't believe there is any mention of WHY the rule of 2 works. If you understand why, you see why the so-called "rule of 4" is just another application of the true rule of 2. Here's the secret: there are 52 cards in the deck, therefore each card represents approximately 2% of the deck. That's where your 2 comes from. The reason this is not exact is that it ignores the card removal effects (there aren't really 52 cards left on the deck at any point you use the rule). This is also why the rule of 2 will always slightly overestimate your odds.

Great post/awesome series DD. wp.

Obviously I can't speak for the FAQ, but this series will be available in the Digest.

When nitpicking probabilities it would help your argument if you used the proper statistical way of calculating rather than stating a flawed way as being correct in your explanation. Probabilities aren't additive and DiamondDog have clearly shown the correct way to make the calculations.

Ans stating there isn't an argument for as why it works when DiamondDog clearly shows the correct way to calculate is disingenuous at best.

Also, stating that the rule of 2 will always slightly overestimate your chances are wrong. You only overestimate your chances when you have many outs when using the rule of 4. The rule of 2 slightly underestimates your chances.

One of the fascinations of Maths (well, for me anyway, although I'm sure I'm not alone in this respect) is that there's nearly always more than one way to arrive at the result.

(Last time I looked there were more than 100 different proofs of Pythagoras' Theorem. Which kind of makes me wonder what on earth was going through the mind of the guy who sat down having seen the first 99, determined to find one more.)

However, if you have a method that works for you, and you understand why it works, stick with it would be my advice.

Like I said in the first post in the series, I'm aiming at beginners - the guys who don't know how to work out this kind of stuff and are looking for a steer.

I'm suggesting they multiply by 2 with one card to come, and to multiply by 4 with two cards to come. You're suggesting they multiply by 2, and then multiply by the number of cards to come. That kinda feels like the same thing to me. What I've described is the rule of '2 and 4'. What you've described might be termed the rule of '2 and 1 or 2' I guess. Either way we're getting the same answers.

I believe that I do explain why multiplying by 2 works, but I guess that comes down to what you accept as an explanation.

chad0x00's post about Solomon's provided a useful adjustment to make the estimates more accurate.

Thanks for the feedback.

Good Luck.

Great articles, they've really helped me better understand more of the math aspect of the game.

One thing i'm not getting is where the 3/11 comes from and how to calculate that. If anyone could explain it i'd appreciate it.

Maybe go back and have a look at the post on Betting Odds where we explain why, if you're getting odds of a:b, you need to be good b/(a+b) of the time.

Good Luck.

Good stuff DD

Hey mods...can you guys sticky this for about a month.

Thanks in advance of this request.

last 2 posts deleted, but wth, thread deserves a bump anyway ...

+1

Great read, thanks for posting.