What is going on?
Your first choice has a 1 in 3 chance of being correct. The other two doors together have a 2 in 3 chance. A host who knows the prize location always removes a losing door from those two. Switching wins exactly when your original choice was wrong.
Work through it
- Choose one of three doors. The prize is placed randomly.
- The host opens an unchosen door that never contains the prize.
- Stay with your original choice or switch to the other closed door.
Put numbers to the idea
Across the three possible prize positions, your original door wins once. Switching wins in the other two cases. In 1,000 simulated games the totals usually sit near 333 stay wins and 667 switch wins, but random variation changes the exact counts.
A little more clarity
Why are the remaining doors not 50–50?
The host’s choice is informed, not a random removal. Your first door keeps its original 1/3 chance; the surviving alternative carries the other 2/3.
Can staying win more often in a short run?
Yes. A strategy can have a lower theoretical chance and still do better in a small random sample. More games make large discrepancies less common.
Are the strategies tested on the same games?
Yes. The batch simulation evaluates both actions against each identical prize placement and first choice, so their wins always add up to the number of games.