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..................................................... The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show Let's Make a Deal and named after its original host, Monty Hall. The problem was originally posed (and solved) in a letter by Steve Selvin to the American Statistician in 1975.
Simulation -
www.mathwarehouse.com/monty-h...
The Monty Hall problem is a well-known probability puzzle named after the host of the popular game show "Let's Make a Deal," Monty Hall. The problem is as follows: You are a contestant on a game show, and you are presented with three closed doors. Behind one of the doors is a valuable prize, such as a car, while the other two doors conceal goats.
The game proceeds as follows: You initially choose one of the three doors, let's say Door 1. Before revealing what is behind the chosen door, the host, Monty Hall, who knows what is behind each door, opens one of the other two doors to reveal a goat. Importantly, Monty always opens a door that you did not choose and that he knows does not contain the prize.
Now, here's where the tricky part comes in: Monty gives you a choice. You can either stick with your original choice (Door 1) or switch to the remaining unopened door (Door 3). What should you do to maximize your chances of winning the prize?
Intuitively, one might think that the probability of the prize being behind either door is 1/2 at this point, since there are only two doors remaining. However, the optimal strategy is to switch doors. The reason behind this counterintuitive result lies in the concept of conditional probability.
When you initially chose Door 1, the probability of it containing the prize was 1/3. This means that the probability of the prize being behind one of the other doors (Doors 2 and 3 combined) was 2/3. When Monty opens Door 2 (revealing a goat), the fact that he never opens the door with the prize behind it provides new information. The probability distribution now changes. The prize is still behind one of the three doors, but Door 2 is no longer in the running.
By switching to Door 3, you effectively combine the remaining probability mass (2/3) onto that door, doubling your chances of winning the prize compared to sticking with your original choice (1/3). This counterintuitive outcome is often misunderstood, but it has been mathematically proven and demonstrated through simulations.
In summary, the Monty Hall problem challenges our intuitive understanding of probability by demonstrating that switching doors after the host reveals a goat maximizes the chances of winning the prize. It highlights the importance of considering conditional probabilities and how new information can affect the probability distribution.
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