Why does the Monty Hall problem fool almost everyone?

In the Monty Hall problem, a contestant picks one of three doors, the host opens a different door that hides no car, and the contestant may then switch. Switching wins two thirds of the time and staying wins one third. Most people instead judge the choice to be 50-50, and researchers report that people stick with their first door around 85 to 90 percent of the time. Studies point to several causes working together: overlooking what the host knows, an assumption that remaining options are equally likely, and emotional biases about switching. [1] [2]

Suppose you pick door 1. If the car is behind door 2, the host must open door 3, so switching wins. If the car is behind door 3, he must open door 2, so switching wins again. Only if the car is behind door 1 does switching lose. Your first pick is right one time in three, so switching is right in the other two cases out of three. Put another way, the host has a free choice only when you picked correctly. When you picked wrong, he has just one door he can open. [1] [2]

A systematic review says people often fail to notice or keep in mind that the host knows where the prize is and never opens that door. They also do not seem to realise that the host's behaviour depends on their own pick. This detail matters. If the host opened one of the two unchosen doors at random, with a real chance of revealing the car, then staying and switching would be equally good. The host's knowledge and constraints are what change the odds. [2] [1]

Researchers link the 50-50 answer to the equiprobability bias, the habit of assuming that all presented outcomes are equally likely. Studies of how people compute probabilities found that they tend to divide the number of prizes by the number of remaining options, which gives equal odds for staying and switching. One review adds that people often represent the setup poorly from the start. In card-game versions, only about 50 percent of undergraduates understood that the two non-selected options hold the ace 2 times out of 3, and a common answer was 1 of 2. [2] [3] [1]

Social psychologist John Petrocelli had participants play repeatedly, and it took many trials for them to learn that switching was better. He attributes this to counterfactual thinking: losing after switching makes the thought of what might have been very salient, while losing after staying put does not. Reviewers also describe a preference for the first choice. They report that repeated practice or high incentives can reduce choice biases, but practice alone does not give people correct probabilistic reasoning, and guided explanations bring only modest improvement. [1] [3]

Walter Herbranson, a comparative psychologist, notes that even people well trained in mathematics take the wrong intuitive approach, which he sees as a sign of a normally functioning human brain. [1]