Rationale for Rule of 4

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nondescript

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Jun 3, 2016
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Hi everyone,

It's my first post here! Hope to be a part of the community. My question concerns the rule of 4/2. I'd like to ask what the logical reason is behind the approximation. I figured out the reason for the rule of 2: the probability that you win in the river given that you are in the turn is = outs/44 = 2 outs / 88 which is approximately 2*outs / 100. You can use the same reasoning if you are in the flop and only want to see whether you'd win if the game stops in the turn because this time the probability of winning is = outs/45 = 2*outs/90 which is again approximately 2*outs / 100. Please correct me if my reasoning has flaws. But I don't understand how to derive the rule of 4.

Can anyone help with this? Thanks!
 
M

makrarom

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Feb 25, 2016
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You got me confused here, friend. I honestly, don't know what you're even asking, but you got my interest, so I'd ask if you could explain it a little better, or if someone else tries to answer his question and help.

I don't understand where did you get /44 number and what does the "outs" stand for? Because if outs are x, how can you determine you're going to have 2 outs? Do you mean outs as one out?

I also don't know what those big numbers stand for (88,100,90..). Please explain, because I'd like to understand :)
 
T

Two6JJ

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Feb 27, 2016
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This was borrowed from the thepokerbank.com's explanation. It seems to get to the ease of the rules intention.

The rule of 4 and 2 is a quick shortcut for helping you to work out the percentage odds of completing a draw in Hold’em. To get your percentage odds:

Multiply your outs by 2 when you are on the flop waiting for the turn.
Multiply your outs by 2 when you are on the turn waiting for the river.
Multiply your outs by 4 when you are on the flop waiting for the river (opponent is all-in).
When you have multiplied your outs by either 4 or 2, you will get a percentage that you can compare with your pot odds to work out whether or not it’s worth calling with a drawing hand.

The rule of 4 and 2 just works for percentages odds, not for ratio odds I'm afraid.

Examples of the 2/4 rule.

Flush draw: 9 outs * 2 = 18%
Straight draw: 8 outs * 2 = 16%
Two overcards: 6 outs * 2 = 12%
Two pair and you need to make a full house: 4 outs * 2 = 8%
Flush draw, opponent is all in on flop: 9 outs * 4 = 36%
Straight draw, opponent is all in on flop: 8 outs * 4 = 32%
 
N

nondescript

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Jun 3, 2016
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Two6JJ, you've given an explanation of the rule, but not the reasoning for it. I've finally figured it out though after looking around the net. I will explain below.

Makrarom, outs here is just my label for the number of outs. It's not that there are 2 outs here, it's that we multiply the number of outs by 2. I explain further later. 44 is gotten by 44=52-2-2-4 as we deduct our cards, our opponent's cards and the flop and turn cards from the possible cards. 88 is just twice 44 (we multiply top and bottom of the fraction outs/44 to get 2*outs/88 as multiply top and bottom of a rational number preserves equality). 100 is just an approximation of 88 in order to write the fraction as approximately a percentage.

Now for the reason for the rule of 4:

The probability of getting the best hand in the river while on the flop is given by:
(outs/45)*(44/44)+(45/45)*(outs/44) = outs/45+outs/44 = 2*outs/90 +2*outs/88 which is approximately 2*outs/100 + 2*outs/100 = 4*outs/100.

We don't make the calculation for backdoor outs and instead count them as non-backdoor, but that should be approximately valid.
 
Nicebrew

Nicebrew

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Jan 31, 2016
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Hi nondescript! I hope I can shed some light on your conundrum!

So, I think firstly you are just misunderstanding the rule so let me highlight where I think you're going wrong.

Your main mistake is that you are deducting the opponents cards from your total number of cards. The idea of the rule is that you can see two cards in your hand and 3 cards on the flop which leaves a total of 47 unseen cards to account for. Since we have no way of knowing exactly what our opponent holds, we still have to leave their cards (whatever they may be) into our calculation. Let me give an example.

if you are holding 9,10 of hearts and the flop comes 4 6 J with two hearts. you know that any heart will complete your flush.

so of the 5 cards we can actually see, 4 of them are hearts, leaving another 9 hearts left on the table. Again, since there's no way of knowing whether our opponent holds any of them, we have to give them all an equal chance of coming on the turn or river. So our calculation is as follows

number of outs (9) X4 (because the rule of 4 and 2 states we times our outs by 4 on the flop) = 36.

We then take this number (36) and make it a percentage which would be 36% of course.

Now this doesn't mean that you have a 36% chance of making a flush on the turn. It simply means you have a 36% chance of making your flush by the river.

The reason for it being 4 and 2 is because on the flop you get to see 2 more cards before the showdown whereas on the turn you only get to see 1 more card before showdown therefore halving the chance that a heart will appear by the river.

To conclude, you are only looking to employ the rule of 4 and 2 when you need ONE more card to make your made hand. you will not be able to use the rule to accurately predict the likelihood of hitting two consecutive outs to make a hand. You are also only including cards that you can SEE into this equation. (flop you will have 5 cards visible, turn will have 6 cards visible).

Try and work out these examples in a reply and I can give feedback.

what is the percentage you will make your hand by the river in the following scenarios;

Open ended straight draw on flop =

Flush draw on the turn =

Set (three of a kind) on the flop =

Gutshot straight draw on the flop =

Flush + open ended straight draw on the flop =

Hope this helps!

Nicebrew.
 
N

nondescript

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Jun 3, 2016
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Thanks Nicebrew! I understand the rule, but I was assuming that I know my opponent's hands. That was why I made such calculations. I am satisfied with the reasoning now.

It doesn't make sense to me that seeing 2 cards vs seeing 1 card is a good reason for the multiplication by 4 and 2 respectively, but the calculation I made makes sense to me. Thanks for the effort!
 
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