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Three questions-also known as Monty Hall questions-come from the American TV game show Let's Make a Deal. The name of the question comes from the program host Monty Hall.
Participants will see three closed doors, one of which has a car behind it. Choose the door with a car behind it to win the car, and there are goats behind the other two doors. When the contestants chose a door but didn't open it, the host opened one of the remaining two doors, revealing one of the goats. Then the host will ask the contestants if they want to change another door that is still closed. The question is: will changing another door increase the chances of the contestants winning the car?
Simple analysis
The answer to the question is yes: when competitors turn to another door instead of keeping the original choice, the chances of winning the car will double. There are three possible situations, all of which have equal possibilities (1/3): the contestant chooses Goat No.1, and the host chooses Goat No.2, and the conversion will win the car. The contestant chooses sheep number two, and the host chooses sheep number one. Modification will win the car. Participants choose cars, and the host chooses either of the two goats. The conversion will fail. In the first two cases, competitors can win the car by changing their choices. The third case is the only case where the contestants keep their original choices and win. Because two of the three situations are won by conversion selection, the probability of winning by conversion selection is 2/3. If there is no initial choice, or the host opens a door casually, or the host only asks whether to change the choice when the player makes some choices, the problem will be different. For example, if the host removes one of the two goats first, and then lets the contestants make a choice, the probability of choice will be 1/2. You can also use reverse thinking to understand this choice. No matter what their initial choice was, when asked by the host whether to change, they all chose to change. If the contestants choose goats first, 100% will win; If the player chooses the car first, 100% loses after the change. The probability of choosing a sheep is 2/3, and the probability of choosing a car is 1/3. Therefore, in any case, the switching option can increase the odds of winning compared with the original car with only 1/3.
Of course, there are also dissidents who put forward 50%
The answer given by people is 2/3, which is of course wrong. How did two thirds come from? Let me explain to you: the probability of the first person choosing a door is 1/3. After the second person opens a door, the car must be in the remaining two doors, so the probability of the last door is 1- 1/3=2/3. This kind of thinking and explanation is completely wrong. What's wrong? "The probability that the first person chooses the first door is 1/3" is wrong. When the number of samples is 3, the probability of 1/3 is obtained. When the second person opens another door, the sample number of the whole event is 2. When the first person doesn't change his choice, he has actually chosen the second time! Just because he chose "unchanged" doesn't mean that his chances are "unchanged". The whole process of the incident is like this: a person chose a door and didn't open it. The other person chooses one of the remaining two doors with a sheep behind it. The first person chooses the first door again among the remaining two doors. So the probability of his choosing a car is 50%.
This is an old question, but it may cause a heated debate at any time. Even more amazing is that both intuitionists and probability theorists think their answers are reasonable. Wittgenstein believes that most problems in the world are ultimately language problems. The debate on the three issues is actually semantic. The correct answer should be: If the host knows in advance which door has goats and he deliberately chooses to open the door with goats, then the contestant should change to another door, which can increase his winning probability from 1/3 to 2/3. If the owner doesn't know in advance which door has a goat or just chooses a door at random, it turns out to be a goat. At this time, the contestants don't need to change doors, and the winning probability is always 1/2. In other words, the root of probability lies in whether it is a man-made event or a purely random mathematical event.