The Monty Hall Problem (= Car-Goat Problem = 三門問題) is a famous probability puzzle based on a game show scenario. ¹ A contestant is presented with three doors: behind one door is a valuable prize (like a car), and behind the other two are goats. After the contestant selects a door, the host—who knows what’s behind each door—opens one of the remaining doors, always revealing a goat. The contestant is then given the choice to stick with their original door or switch (換？還是不換？) to the other unopened door. Counterintuitively, the best strategy is to always switch, as it gives a 2/3 chance of winning compared to a 1/3 chance if staying. The problem illustrates how human intuition about probability can be misleading and highlights the importance of conditional probability in decision-making.

為何換門更好 

• 第一次選擇：你選擇到汽車的機率是 1/3；選到山羊的機率是 2/3。
• 主持人介入：主持人永遠會打開一扇有山羊的門，這個行動是基於你的第一次選擇。
  • 如果你第一次選擇到的是汽車（機率為 1/3），主持人會隨機打開另一扇有山羊的門。這時，換門一定會讓你輸掉汽車。
  • 如果你第一次選擇到的是山羊（機率為 2/3），主持人就只能打開另一扇有山羊的門。這時，剩下未被打開的門後面一定是汽車，換門一定會讓你贏得汽車。
• 結論：由於你第一次選擇到山羊的機率是 2/3，因此換門贏得汽車的機率是 2/3，而堅持原選擇的機率是 1/3。