Monte Hall Problem in Holdem

The Monte Hall Problem is a very popular problem that is confusing to many people. The problem is often misrepresented or misinterpreted by the one presenting the question. Many times, the confusion is due to the person asking the question not phrasing the question correctly or not providing all of the pertinent information. Here is my attempt to explain it succinctly, and apply it to a Hold'em situation.

The setup to the Monte Hall problem is that you are a contestant in a game show and Monte Hall is the show host. He will give you a product (like an electronic device), but then offer you a choice to exchange the product. Usually the result of the deal that he offers you is unclear. You have to make a decision with uncertainty. If you choose to make the deal, then you will have to play a game (it could be as simple as choosing between what is behind door A or door B). If you get lucky, you may wind up with a much better product or nothing at all. What makes the show interesting is that the contestants are sometimes asked to make decisions that are tough and counterintuitive. The famous Monte Hall problem is one of those that are counterintuitive.

In the problem, there are three doors. Behind one of the doors is a brand new car, behind the other two doors are goats. The contestant does not know which objects are behind each door, but presumably the contestant would prefer to win the new car over a goat. The contestant is asked to choose a door, and the object behind the chosen door now belongs to him. The game is not over though. Before Monte Hall reveals what is behind the contestant's door, he now opens one of the other doors to reveal a goat. Then he says to the contestant "You may choose to switch to the other unopened door if you wish." If the contestant switches, does he increase his chances of winning a car?

As is often the answer to the poker question of "How should I play this hand?", the answer to the Monte Hall problem as it is presented above is "It depends." If it was presented slightly differently, the answer would be clearer. The difference is whether you knew ahead of time that Monte Hall would always open up a door with a goat behind it and offer you to switch to the third door. This is an important fact and makes a big difference in the problem. Oftentimes, when others present the problem, they do not make it clear if this is true or not.

If you knew in advance that Monte Hall is 100% guaranteed to open up a door and show you a goat and offer you to switch after you have already picked a door, then the answer is yes, you should switch.

If you were not sure in advance if Monte Hall would offer you a deal, then you simply don't know. Is he bluffing by offering you to switch because he knows you have picked the car? Did he already plan to show you a goat behind one door and offer you the contents behind the other door even before you made the first choice? What is your opinion of Monte Hall as a person, is he nefarious and wants you to get the goat, or is he generous and wants you to increase your chances of winning the car? What is your opinion of the intelligence of Monte Hall? Is he smart enough to realize that if he always shows you a goat and offers you to switch doors that he is increasing your chances of winning the car or does he think that the choice does not matter?

Let's assume the scenario where it is 100% guaranteed that Monte Hall knows what is behind each door and will always show you a goat behind one of the doors whether or not you picked the door with the car behind it. Then he will offer you to switch to the third door that is not yet open. Given that this is the case, then Monte Hall will always show you a door with a goat behind it, whether or not the door you have chosen has a car or a goat behind it. 1/3 of the time, the first door you chose will have a car behind it. That means 2/3 of the time, the car is behind one of the other doors.

Since you knew that Monte will always open up one of the doors with a goat behind it that means if you switch, you will go from a 1/3 chance to a 2/3 chance.

One of the problems with this answer is that many people confuse this issue and think the chances are ^ for both the door that you originally picked and the remaining door since there are only two unopened doors left. They neglect the fact that if the car was behind one of the doors that you did not originally choose, then Monte Hall would not open that door because he knows that the car is behind it, and he is not going to show you the car. Monte would only open the door with the goat behind it if the other door had the car behind it. Thus 2/3 of the time (because 2/3 of the time you have picked the incorrect door, and the car is behind one of the other two doors), Monte has no choice but to open the one door that has the goat behind it. The other 1/3 of the time, he can randomly decide which door he wants to open. This means that 2/3 of the time, the door that Monte did not open will have the car behind it, and the other 1/3 of the time there will be a goat behind the other door because you already picked the correct door with the car behind it. So if we knew with 100% certainty that Monte Hall would open up a door with a goat behind it and offer us to switch cars, whether we chose a door with the goat behind it or we chose a door with the car behind it, then we will increase our chances of winning the car from 1/3 to 2/3, a very significant improvement.

 Door / Contents If you chose door A, Monte will show you: If you chose door B, Monte will show you: If you chose door C, Monte will show you: A - Car B - Goat either B or C B C - Goat either B or C C If you switch to the 3rd door You will get a goat You will get the car You will get the car

However, what happens if Monte Hall sometimes does not offer you the choice of switching? What if sometimes Monte will simply open up the door that you chose? If that is the case, then you need to know that if he offers you to switch doors, then maybe he is trying to bait you to switch to a worse door and away from the door with the car behind it. Maybe Monte does not want you to win the car and will only ask if you would like to switch doors if you actually chose the door with the car behind it. In order to actually take Monte's offer and switch doors and think you are going from a 1/3 chance to a 2/3 chance, you have to be incredibly confident that he will always ask if you want to switch no matter what you chose, and that he does not have a nefarious reason to ask you if you want to switch. This is incredibly important and pertinent information to the problem, and yet many people who present the problem fail to make the facts crystal clear. The problem is already a difficult one to understand conceptually, and the omission of this important fact merely makes it even more confusing to many.

How does the understanding of this problem apply to Limit Hold'em? Well, there are some situations where your opponent is guaranteed to bet. Given certain opponents and certain situations, you may be 100% certain that your opponent will bet. If you know that your opponent is 100% to bet, then that means his bet has no bearings on the quality of his hand. You cannot base any of your reasoning of why your opponent has bet on the latest new piece of information. Of course, you may need to readjust the value of your own holdings based on the new information. The key is that you cannot adjust the value of your opponents holdings based on the new information since he would have acted identically no matter what the new piece of information was. This situation typically presents itself more often in shorthanded games than in full games. Just like in the Monte Hall problem, where you knew Monte would show you a door with a goat behind it and ask if you wanted to switch, you knew that the chances you picked the correct door was 1/3, and it is still 1/3 even after Monte opened one of the doors for you.

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