Behind a car (Game Theory tricks)

In a game program, the host points you to the three doors marked with "1", "2", and "3", and clearly tells you that the two of them are behind goats, there is a famous car behind the other door. You need to select one of the three doors and get the prize behind the selected door. Of course you want to select a car instead of a goat. Since it is three choices, it is clear that the chance of choosing a car is 1/3.

 

Without any help, you chose one (for example, Gate 1), which is neither right nor wrong. It is just a matter of luck. But the host did not open door l immediately, but opened door 3. A sheep appeared behind the door. Now, the host asks if you want to change your mind and choose Gate 2. Now you are faced with a decision-making problem: change or not.

 

This question was raised in an article by the American columnist Ms. safeat. Her idea is roughly as follows: If you choose door 1, you will have a 1/3 chance to get a car, but there will also be a 2/3 chance. The car is behind the other two doors. Then, the good host asked you to confirm that the car is not behind door 3, but the chance of having a car at door l remains unchanged, and the chance of having a car after door 2 becomes 2/3. In fact, the probability of door 3 is transferred to door 2, so of course you should choose another one.

 

Sequent's game attracted thousands of readers, most of whom thought that her inference was wrong, and claimed that the first and second doors should have the same chance, the reason is that you have changed your choice to 2 and 1, and you do not know which door has a car behind it. Therefore, the chances are the same as throwing a copper coin.

 

Interestingly, sevante found another interesting phenomenon: In a general public letter, 90% thought she was wrong, and in a letter sent from a university, only 60% opposed her opinion. In the follow-up development, some statistical doctors joined the discussion and thought that the probability should be 1/2. Saifant was surprised by the craze and opposition caused by this issue, but she insisted on her own opinions.

 

Statisticians have been seeking answers to these questions from the past and today. In fact, it is simple. Everyone can understand it or verify it in person. Here we can simulate: use three cards as the door, one a and two Ghost cards as the car and goat respectively, and play for more than a dozen times in a row.

 

You will soon be able to find that changing a card is more advantageous, just as Saif said. Why are these experts still arguing? Why are the chances of the first and second doors not equal after the goat appears on Gate 3, or do all gamers have some unknown assumptions, that is, using playing cards for simulation?

 

A fair game, so the initial probability of each game is 1/3, so far no problem.

 

Now you have chosen Gate 1, and there is no problem here, because you do not know anything, so the chance to guess is 1/3.

 

The key part is that the host Opens Door 3 without explaining why he wants to open door 3. There are several possibilities.

 

The host may only want to play tickets. As long as the player chooses No. 1, he must open door 3. no matter whether the door is behind door 3 or not. If the door appears, it is lucky; if it is a car. Then the game will come to an end, and the players will lose. If the host really thinks so, it is not a new news for you because the third door is not a car. At this time, the car may be one of the first or second doors, there is no special preference between the two. The host did not give you a good reason to change the door, nor did you provide reasons to keep your case.

 

Most of the opponents of safeat believe that under such circumstances, the probability is equal, but they do not know that they have assumed the host's strategy.

 

However, if the host has another set of rules, he knows that he cannot open the door with a car, because it will damage the suspense atmosphere at the scene. The early end of the game will make the audience lose interest. The host who takes the entertainment masses as its duty should be its firm target to attract the audience.

 

Therefore, if the host's strategy is never to open the door with a car, then if you choose the right door from the beginning, he can open door 2 or door 3 at will; if you choose the wrong one from the beginning, then he will open the door without a car. So in any case, the door he opened must be a goat, so there will be no new information.

 

Under such circumstances, no matter where the car is, his behavior will not affect the initial choice, that is, the probability of the first door. If the car is not behind Door 1, the door he opens will tell you where the grand prize is located. Therefore, Door 2 has a 2/3 chance. If you choose door 1 for the first time, you will be wrong, he has already told you which door to choose. If this is the host's policy, you will be able to change the name of the car. Although you may not be sure that you will win, you may still have the right chance to choose the first option, but the chance to win is doubled.

 

Because the assumptions made to the host are different. Therefore, both parties may be right. Assuming that the host opens the door randomly and the car is not behind the door opened by him, the chances are 50%. Assuming that he has long decided not to open the door with a car at this stage, he asked you to check what is behind door 3, and you should use this information for another choice.

 

When you are at a disadvantage in the game, it is advantageous to take a bigger risk to change the card. When you are in a favorable position, it is wise to adopt a conservative strategy and play cards with the other party.

 

Note: My understanding of this issue is that the host aims to create suspense entertainment for the masses, so his strategy should be to avoid opening the door with a car, assume that the car is behind Door 1 and the probability is 1/3. The host can open door 2 or door 3 at will. Suppose the car is not behind Door 1 and the probability is 2/3. Then the host will not open the door with the car or Door 1 (because you chose door 1, if it is opened, it will lose the suspense.) He will open the door without a car in door 2 and 3. If you change your strategy, the chance of winning the car will increase to 2/3.