Why does switching give you 2/3?
Your first pick has a 1/3 chance of hiding the car; the other two doors together have 2/3. Because the host knows where the car is and always opens a losing door, that whole 2/3 collapses onto the one remaining door. List the cases: in the two where you first picked a goat, switching wins; only when you picked the car does switching lose.
Why does everyone think it's 50/50?
Two doors left feels like a coin flip, but that ignores the information in the host's choice. Imagine 100 doors where the host opens 98 losers—switching suddenly seems obvious. When Marilyn vos Savant told readers to switch in 1990, she received thousands of letters, many from PhDs, insisting she was wrong. Simulations proved her right, and Bayes' theorem gives the same 2/3.
FAQ
- What if the host opens a door at random?
- If he doesn't know and happens to reveal a goat, switching and staying are both 1/2.
- How can I check it myself?
- Play it out with three cards a few dozen times, or run a quick simulation—it converges near 2/3.
Sources reviewed: 2026-10-25
References
- Selvin, S. (1975). A problem in probability [Letter]. The American Statistician, 29(1), 67.
- vos Savant, M. (1990). Ask Marilyn. Parade Magazine, Sept. 9.
- Krauss, S., & Wang, X. T. (2003). The psychology of the Monty Hall problem. Journal of Experimental Psychology: General, 132(1), 3–22.