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The Monty Hall problem is perhaps the most famous probability puzzle ever posed. You are presented with three doors. Behind one is a prize; behind the others, nothing. You choose a door. The host, who knows what is behind each door, opens one of the remaining doors to reveal nothing, then asks if you want to switch. Should you? The counterintuitive answer is yes — switching gives you a 2/3 chance of winning, while staying gives only 1/3. Many brilliant mathematicians initially disagreed with this result, including Paul Erdős. In this thread, we explain the solution using multiple approaches: tree diagrams, conditional probability formulas, and simulation results. We also explore why the problem is so psychologically difficult. If you have a variation or a related puzzle, share it here for the community to solve together.
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