The monty hall problem in opening ranges

The monty hall problem ( is a cool statistical phenomenon in which you are on a game show and given a choice of 3 doors, one of which has a car behind it. You choose door 1 and the host shows you that door 2 does not contain the car and asks if you’d like to stick with your choice or switch to door 3. Should you switch? The answer is yes. Many people believe that the probability is the same for doors 1 and 3, but in fact additional information has been given and there is a 2/3 chance the car is behind door 3. If you want more info, check out the wikipedia article.

So how does this apply to poker? Say you are sitting on the dealer button. Every player before you gives you information based on how they act. Let’s say you are playing full ring and all the players fold before your button and you have to decide whether to open J9o. Now compare that to if you were playing 3 handed. It might seem like the blinds have completely random hands, but you actually have a lot more information in full ring (the 12 cards your other opponents folded), which makes it more likely that one of the blinds has a stronger hand. I have noticed from playing that when it folds around one of the remaining players frequently has a monster and this makes sense given the bad cards held by earlier players. Should opening ranges be adjusted based on game size? I would much rather open J9o from the button 3-handed than after it folds around to me in full ring. Or is the effect too small to be meaningful? Just a thought…


If there is an effect, I say it is meaningful. Everything counts in large amounts.

It does feel like it could make a difference, but I have never factored it in.

If 6 players folded before our turn, they would have had to have folded exactly no aces to make the hole cards of the blinds better, though?

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Nice post @JoeDirk - I’m not sure we can gather much information from the hands that were folded before we acted unless we have players opening every Ax or something like that. If players are using pretty standard ranges, I don’t think we can assume much based on EP/MP non-action. UTG could very well have a hand like A8o, which he is not opening but he did burn an ace. You could have a string of non-playable hands from earlier positions that contain many Ax and Kx combos. Its possible that all aces or kings could have been mucked by the time the action gets to you on the button if the MP players are still not opening off-suit A7, K6 etc.

Looking at it the other way, we certainly do gain information when hands are opened before we have the chance to act. I think this is the standard way for players to gage whether to play certain hands or not. An UTG open has many premium Ax hands so we don’t look to get in the action with our easily dominated aces when facing an open from that position. Where we could comfortably 3-bet a CO open when holding AJo on the button, we really couldn’t profitably 3-bet an UTG open with that same hand from the same position.

Good food for thought but I don’t think folds reveal enough information to be useful, or at least not as much information as opens/flats do.

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Folds definitely don’t reveal anywhere near as much information as calls or bets, but particularly on replay or in other games with a lot of limping I do expect many players to at least limp any Ax. It is certainly less useful to look at folds in a game full of good players using better-informed ranges and not limping.

Also the responses have focused on Aces and kings, and while they do comprise a large proportion of “good” hands, in a game as loose as replay, players are likely to at least limp any pocket pair or suited Broadway. It has to at least marginally increase the amount of information, not in any particular situation but on average, that it is less likely that Broadway cards or pocket pairs have been dealt to earlier positions. I wonder how the probability of being dealt a pocket pair is impacted when given the knowledge that no previous player has been dealt one. Seems like it would be pretty marginal and not super useful given that it may be a poor assumption, but fun to think about.

Nice problem. Initially every room has a probability of having the car as 1/3rd. I choose a room, the other two rooms have the probabiliy 2/3rd. Host points out one room that doesn’t have the car, so the remaining room has 2/3rd probability. So its better switch to that room which has 2/3rd probability.

Looking at the other way, I chose one room, and the host points out one room which doesn’t have the car. Now I have to choose between the other two rooms, which is a 50/50 probability. What’s the mistake I am doing here, I am ignoring the fact that host pointed out a room after I have chosen the room.

So, properly integrating different random events produces different expectations. (Integrating non random events also produces different solutions, like bending strength of an ordinary beam and a c-beam).

With the same argument one can say game of roulette can be beaten. Based on the argument, a pattern (red,odd,above/below) does not repeat indefinitely. I wrote a small program that produces random number between 1 and 36, and 0 and 00, and checked the repeat of above or below 18 pattern. 0 and 00 are considered repeat. It didn’t exceed 27 for a million run. So within the limit of 27, if I double my bet each game on the opposite of the pattern until the pattern breaks, I can beat the game of roulette. Its not possible practically , I need a bank roll of 67 million chips to win 1 chip for each breaking of a pattern, but mathematically it argues.

To extent to poker, if I don’t win say twelve hands in six player table, sooner I should get a winning hand.

I don’t know whether any work done on a integral of random events and their limits, but its a compelling thought.

I don’t see how the fact that everyone has folded around makes a hand like a pocket pair more likely in the remaining hands. There’s roughly a 6% chance of getting a pair, but that doesn’t imply that a pair will be out 54% of the time at a 9 seat table. If one flips a coin and calls “heads” they should be right 50% of the time, but this doesn’t mean it will be heads at least once 100% of the time if they do it twice. With pocket pairs at a full table, you have nine 6% chances, not one 54% chance.

In the Monty Hall problem, you are getting 100% reliable information, so yes, it does change the odds. There could be any of a number of reasons people fold small pairs of AXs type hands, even if they would usually limp them. So here we aren’t really talking about information, but rather the quality of the information.

Does unreliable information have any value at all?

i once thought of the same thing as well, and checked the internet if such a strategy exists. it does and it’s called the martingale system.
it indeed requires you to double your bet each time to eventually make profit. the biggest flaw is also like u said, that eventually when you get the inevitable bad luck streak, it will require tons of money to win just 1 chip. but anyway it’s a fun strategy.

It’s a much weaker effect than the Monty Hall situation, and I am not even sure there is a tangible difference. Also, I am not saying there would be a greater likelihood of a pocket pair because the distribution of the cards to the previous players is random. But you could assume (especially on Replay) that if it folds to you on the button, no 1 player had two 9s or two 10s etc, and even that no aces have been dealt or no two broadway cards have been dealt to one player, and while each of these assumptions may be small (because the other players may have folded A6o, A3o, K8o, Q7s, K4s, J5s, accounting for a large number of broadway cards), it could possibly be more likely that one of the blinds is holding an ace or even a “big hand” in general because you know that big hands have not been dealt to any other player, even if the components of those hands could have been distributed to different players.

Maybe a 1 or 2% edge makes a difference in ring games. I’m a tournament player, and such a small edge is all but useless to me, and i don’t have that good a read on most of the players I see.

It probably does have some effect on the game, but I think it’s hard to make practical use of such questionable information.

Edited to add: Aren’t you basically trying to quantify the advantages/disadvantages of position? Yeah, of course being in late position is an advantage, but how much exactly? That’s a question with too many variables to quantify, at least in any meaningful way.

Yes, folds contain information, even if the information that they contain is weaker. Taking a Bayesian perspective, any information is worth updating on.

This thread is relevant to your musings. For instance, the third post claims that after observing six folds in a full ring, we expect the remaining cards to be drawn from a deck that is richer in aces and kings while being scarcer in deuces and treys. The numbers computed by whosnext shows that in those twelve folded cards, we expect 0.5 aces and 1.1 deuces.

You are correct in your inference that as a result, the hands in the blinds become progressively stronger after each fold. The logical conclusion is that button opening ranges in 3-handed play should be looser than button opening ranges in full ring.

To further exemplify this effect, post #18 in that thread shows that there is a 0.59% chance of the big blind having aces when folded to in 6-max. This probability increases to 0.69% when folded to in full ring. Thus, the big blind is 17% more likely to have AA when walked to on a larger table, despite that fact that a priori both hands were random.

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Nevermind. I guess you’re saying 0.69 is 17% more than 0.59%. This is no doubt true, but is it enough to change the way you play the hand?

I would look at it as 0.1% more likely, which is meaningless to me.

Those small differences is apparently enough for a GTO solver to slightly tighten its 9 max button opening range compared to a 6 max button open.

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Thanks for your post and the link.

Quoting from the article
" The fundamental reason why all martingale-type betting systems fail is that no amount of information about the results of past bets can be used to predict the results of a future bet with accuracy better than chance. In mathematical terminology, this corresponds to the assumption that the win-loss outcomes of each bet are independent and identically distributed random variables an assumption which is valid in many realistic situations."

This is the typical argument everywhere given and generally discard considering the possibility of integration of random numbers within certain limit may give different patterns/expectations. I am wondering whether such thing is there.

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Well, yeah, I think most people have a wider open range 6 handed vs 9 handed. I still don’t think the Monty Hall thing applies here though.