We all know that calculating the chance of improving your hand on the turn and/or river is essential information for good poker performance. Of course, you can use software like PokerStove to calculate exactly the chances of improving your hand. But in practice, you need to do this calculation in your head and quickly. But how?
Let's analyze a game situation.
Practical Example:
Blinds at 10/20. You are in the big blind with :As :5c . The whole table folds to the small blind, a solid and aggressive player, who raises to 60 chips (3BB). You call. Pot at 120 chips.
Flop: :Jc :4d :3h
The villain, first to speak, bets 1/2 of the pot (60 chips). What now?
The 4-2 Rule
An excellent approximation for the odds of improving a hand can be calculated using the Rule 4-2:
Count the number of outs (cards that improve your hand) you have. If you are on the flop, multiply the number of outs by 4; if it is on the turn, multiply by 2 the number of outs. The result is the approximate probability (in %) of improving the hand.
Going back to the example, let's assume that your opponent, a solid player, has made a high pair (let's assume he has JQ or KJ). To beat him, you will need to hit an Ace (there are 3 left in the deck, so 3 outs) or a 2 (to make a straight, 4 outs). Therefore, you have 7 outs. Multiplying 7 by 4, we have 28, that is, you have about 28% to improve your hand by the end of the hand.
Since your opponent bet 60 chips, the pot has 180 chips and you need to call 60 for a final pot of 240, i.e. 60/180+60 = 1/4 = 0.25 = 25%, so the call is mathematically correct*. You call.
*Why is this mathematically correct? If you have a 28% chance of winning the pot, and you have to pay 25% of your value to continue, it means that in the long run you will make a profit in situations like these. Remember this concept!
Turn: :5s
You have hit a pair, but you are still losing to a higher pair. You still have the 7 outs from before, but you have two more outs, since a set of 5s would have won you the hand. With 9 outs, by the rule above, you make 9 times 2 (since it is on the turn), which equals 18, so you have about a 18% chance of improving your hand on the river.
The opponent bets half the pot (120 chips) again. Once again you would have to call at a rate of 25% (120 for a final pot of 480), but now your odds of winning are about 18%. Folding would be the mathematically correct move (always remembering that many other variables must be taken into account when deciding whether to fold, such as the villain's ability to bluff or the chance that the villain will fold the hand if a raise is made, but this is another more advanced discussion).
For comparison purposes, making the exact calculation with statistical programs (considering the opponent's hand as :Js :Qd), the probability of improving by adding the turn and river is 29% (using the rule we found 28%), and with the :5s on the turn (with only the river missing) it is 20% (with the rule 18%).
It's an easy method that helps a lot in situations of doubt. Apply it in your daily life and stop chasing pots when it's not mathematically correct to do so!
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very good !!!!
Congratulations Petrillo, this was exactly the subject we discussed on Skype and I see that you fulfilled your promise exactly... now I understand and I will be using this at the tables... oh watch out for me, I will be even more competitive... lol
Petrillo, I just have one observation regarding your article. Explain more what the rule of 4 means. A beginner will tend to think from your explanation that with 7 outs on the flop there is approximately a 28% chance of hitting on the turn, which is not true. These 28% refer to the chances of hitting one of these outs by the river, that is, another bet can still be made, which makes the calculations not so simple. To find out the chance of hitting your outs on the turn, use the 2+1 rule, the statistic is very similar to that of the river. Ex: 7 outs on the flop, the chance of hitting your card on the turn is 7×2 = 14 + 1 = approximately 15% (real chance = 15.55%).
Good people, this post was later calculated by programs, it will hardly go wrong, since those who watch poker on TV see this exact probability, I see it directly in High Stake, on the turn the probability drops by half, or less.
Thank you very much TSawyer and Bruno!!
Bruno, your comment is perfect. I actually didn't forget this detail, I just considered it advanced. So, it will be in Part 3: Advanced Concepts.
And that's right, the probability of improving your hand is up until the river when you're on the flop, the number of outs is 4. If you only count the turn, it's also 2 outs.
OK, Petrillo! I was sure you knew that, I just wasn't sure if you had forgotten to comment or if you would leave it for the next parts. Excellent article! This type of article is very good because it attracts a lot of people to pokerdicas. Who has never gone to Google and typed in some concept about poker that they wanted to know? Having this content here on the site gives more visibility to POKERDICAS!
Guys, thanks for the kind words about the article. We will soon publish part 2. Stay tuned!
Good article, objective and clear. Remember that this rule applies well to few outs, considering the optimistic model (none of your outs are in the opponent's hand). I don't know why some authors prefer this model to considering the opponent's 2 cards as random.
Obviously, the opponent's cards are not completely random, you can stipulate which ones they are, but in a flush draw, for example, can you tell if the opponent is not holding one or two cards of that suit?
Just to think about. The difference in the end is not that high (from 2% to 14 outs), it hardly makes a difference at the game table. What really counts is the comment above, about not forgetting the bet after the turn.
Petrillo great article!
Really very good, this really helps when betting.
I would like to know if you have any other article informing about the probability of winning with a certain hand according to other cards, ow if you know of any site or place that can help me I would be very grateful, thank you
What is the proportion between the pot/call calculation and my probability of winning, to at LEAST accept the play???
If my probability of winning is 70% for example and the pot/call is 72%, should I call or fold???
grateful
ooou petrillo.. putss and I got all tangled up trying to follow the train of thought and pans..
Do you have an email or something so I can explain it to you properly :S
thanks
I couldn't understand this part about “outs”, I didn't find the article very clear.
Hi Elton, I edited the article to make it clearer, thanks for the tip.
[…] Rule 4-2 (Part 1): Basic Concepts * Bankroll Management: The Ladder to High […]
Very good, even though I already know these articles I went a little deeper…
Thank you Marcelo PD!!
In this situation I like to raise to 120 after the flop to see the small player's attitude.
It is mathematically incorrect (if my calculations are correct) to put 120 and a pot of 360 (33%) with a chance of 28, 29% but I think it is good because it is important information that avoids a greater loss on the river.
What do you think?
Congratulations on the blog!! Very good!
If it is mathematically incorrect, and your opponent calls your bet, it will cost you in the long run.
I would like to know more details about the 4-2 rule because I am getting confused about the outs because I would necessarily have to count on the flop cards or just the cards that would make my hand better?
Those that are already on the table are not counted. Only those that are yet to come are of interest!
If more players call, does the calculation remain the same?
Yes, your chance of improving your hand theoretically doesn't change. But if you call a small amount in a big pot it will usually pay off to chase your draws.
Hello! I recently had a great experience and would like to hear your comments, experts, because in my opinion this rule was broken…
Look at the details of the hand:
My hand: 8s 8d
Villain: Jh 2h
Flop: 4h – 5s – 6h
I go all in, the villain pays and says: I had to pay, I had 52%.
Great, it turns out he actually had 16 outs…. (flush=9 outs, straight=4outs, parJ=3 outs)
using the proposed rule we have: 16 outs x 4 + 1 = 65%
Putting these numbers into PokerStove gave 52%, the same probability that the villain calculated at the time…….
WHERE IS THE ERROR!?
ps: in the end the villain won with a JJ on the river…
Renã, you counted 4 outs for a straight when in fact there are 3 (one was already counted for the flush). That's 15 outs, which by the rule would be 60% until the river. It turns out that calculations for more than 10 outs don't follow this rule so well, you have to adjust downwards a little.
This is because you can hit his card and still lose. For example, a “J” comes and then an “8”, you win. Or the 8h comes, he makes a flush, but then doubles a card on the river and you complete the full.
So the tip is that with more than 10 outs you have to reduce the calculation.
It seems that you need to eliminate 2 more Outs by 8h, of the 15 Outs you mentioned for counting Flush and Straight Outs, since 8h would favor Renã's hand. In this way, there would be a total of 13 Outs x 4 = 52%
Was I wrong?
Excellent article.
I've never done calculations this way before and I've been playing for a while now.
Continuation.
Cool Marcelo, what surprised me was the fact that he calculated this on the spot and so on... is there any other rule for calculating the probability?
It is impossible for him to calculate correctly at the time.
You see, he wasn't sure that the pair of J's would win, nor that his flush would win. He could only guess at these things. I think he must have only considered 13 outs, and by his calculation he came up with 52%.
The “mathematically” correct call, I think, is wrong, because you considered the outs of the turn and river but only considered the bet on the turn. Not to mention that you assume that the villain only has KJ or QJ, disregarding the Ace that the villain could have and take a much bigger pot if the hero thinks the game is won… I think that in any case this would be an easy fold.
Likewise, if your card hits, more money can go into the pot. The 4-2 rule is just an estimate, and should be evaluated on a case-by-case basis.
Regarding the article, I had a question:
How to calculate the number of outs if there are two cards missing from a straight?
Example: I have 6 and 7 in my hand, and the flop comes 2, K and 9.
For me to complete a Straight, I would have to get 8 and 10, or 5 and 8. How many Outs would I have to use in the first calculation?
Regarding the article, it was great! Congratulations!
In this case, you should practically ignore this hypothesis. The chance is 24% for one of the right cards to come on the turn, plus 8% for the river to come. Using a statistical calculation, we arrive at a total chance of approximately 2%. It will hardly be worth making a decision counting on these 2%.
Dude, you explained it right, but over time playing poker you will see that outs are not always inevitable, in this case you will have to be sure that the villain has something like KJ or QJ, unless you are sure, and in poker we never have 100% for sure, I would only pull the outs to the nuts in the case of any 2, giving 4 outs.
In your example you are having pot odds of 4 – 1 which is wrong for 7 outs, for your EV to be positive the pot on the flop should have at least 5.5 – 1:
E.V = (7/47 x 240) + (40/47 x – 60)
E. v = (0.14 x 240) + (0.85 x – 60)
EV = (33.6) + (-51)
E. V = – 17.4 in the long run you are losing…
I don't understand the part about paying 60 to win 240. I have to have 60 chips to capture the 180 in the pot, that is 33%. If my odds are 28%, I don't have pot odds. If I have to put in 60 to win the 60 I just put in, I shouldn't consider them. Am I wrong?
Dante, the calculation is always based on the final pot. The pot does not represent my “profit”, only how many chips I have.
So for the calculation, you do what you need to pay divided by the final pot (what was in the pot plus the value of your call).
In this case, it is 60/(180+60).
Another way of doing the math, very common in the US, is to say that you have odds of 3:1 (180:60), which represents 25%.
The 33% percentage is when I have odds of 2:1.
Thanks! 😀
Thanks for the clarification Petrillo.
Hi. I didn't understand the result of the pot calculation. 60/180 + 60
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