Monty Hall

Definition The Monty Hall problem is a probability puzzle about a game with three closed doors. One door hides a car and the other two hide goats, and the car is equally likely to be behind each door, so each door has probability $1/3$. The player picks one door. The host knows where the car is and follows three rules. He always opens one of the two doors the player did not pick, the door he opens always has a goat behind it, and he always offers the player the chance to switch to the other closed door. When the player’s door hides the car, both other doors hide goats, and the host chooses between them with probability $1/2$ each. The player then stays or switches and wins whatever is behind the final door. Under these rules staying wins the car with probability $1/3$ and switching wins it with probability $2/3$. The same holds after the player sees which door was opened. For example, if the player picks door 1 and the host opens door 3, the car is behind door 2 with probability $2/3$. The answer comes from the host’s rules, not from the number of closed doors, and a host who follows different rules can make switching win with a different probability.

Definition The sample space of a random experiment is the set of all its possible outcomes, described in enough detail that every run of the experiment produces exactly one of them. An event is a set of outcomes. When the sample space is finite, each outcome has a probability, the probabilities are zero or positive and add up to $1$, and the probability of an event is the sum of the probabilities of its outcomes. An event made of $k$ outcomes has probability $k/N$ only when all $N$ outcomes are equally likely.

Definition For events $A$ and $B$ with $P(B) > 0$, the conditional probability of $A$ given $B$ is

where $A \cap B$ is the event that $A$ and $B$ both happen. It is the probability of $A$ once we know that $B$ happened. To compute it, keep only the outcomes in $B$ and divide their probabilities by $P(B)$, so that they add up to $1$. The event $B$ has to describe exactly what was learned, in the sample space of the experiment that was actually run. The lesson Conditional Probability in the probability course develops the idea further.

Definition In the Monty Hall problem the switching strategy is the plan, made before the game starts, to accept the host’s offer every time and move to the other closed door. The staying strategy keeps the first pick every time. Under the host’s rules exactly one of the two strategies wins each game, and switching wins exactly when the first pick hides a goat. Hence switching wins with probability $2/3$ and staying with probability $1/3$, whichever door the player picks first and whatever rule the host uses to choose between two goat doors. More generally, take $n \ge 3$ doors, one car placed at random, and a host who knows where the car is and always opens $n - 2$ of the unpicked doors, all with goats, before offering the one unpicked door left closed. Then switching wins with probability $(n - 1)/n$. A higher probability does not promise a win in any one game. If the host does not know where the car is, or offers the switch only in some games, these probabilities can change.