If you guess on a 3-door game show, should you switch doors after a hint?
After you watchIf you guess on a 3-door game show, should you switch doors after a hint?
The short answer
Yes — you should switch. On a 3-door game show where the host knowingly opens a losing door, switching wins about 2 times out of 3, while staying wins only about 1 time out of 3. Switching nearly doubles your chance.
Try this next
- What if you only ran 10 rounds instead of hundreds? Run a short batch a few times and watch the switch line. Predict first: will it land near 2-in-3 every time, or bounce all over? See how few rounds makes the answer look like luck.
- What if you always stayed instead of always switching? Pick the stay strategy and predict where its line will settle before you run. Watch it crawl toward 1-in-3 instead of 2-in-3.
Now you — bend it
- What if What if you bumped the game up to 10 doors and the host opened 8 losing ones?Your first pick is now just 1-in-10, so almost all the chance piles onto the one other door left. Predict how lopsided the switch line gets before you run it.
- What if What if the host opened a door at random and sometimes showed the prize?Now the host's hint is not loaded anymore. Think about what happens to the leftover two doors when nobody is steering away from the prize.
Can you prove it?Switching wins about 2 times out of 3, not 1 in 2. — Run hundreds of rounds on the switch strategy and watch the win-rate line settle near 2-in-3, then run the stay strategy and watch it settle near 1-in-3. The gap is real and it shows up every time.
Design your own test:Predict how many rounds it takes before the line stops bouncing and settles — then run small batches and big batches to test your guess.
Explain it to a 6-year-old: Your first cup is usually wrong, so jumping to the other one the helper left closed wins more often.
The whole story
How it works
Your first pick is just a 1-in-3 guess, so two times out of three the prize is behind a door you did not pick. The host knows where the prize is and always opens a losing door, never the prize. Because the host's choice is loaded and never lands on your door, your first pick stays at 1-in-3 — and the whole leftover 2-in-3 chance piles onto the single other closed door. Jumping to that door means you win whenever your first guess was wrong, which is two times out of three.
What people get wrong
It feels like once one losing door is open and two doors are left, it must be a fresh 50/50 coin flip, so switching cannot matter. That is the trap. The two doors are not equal: the host deliberately avoided the prize when choosing what to open, so the door you skipped carries the full 2-in-3 chance and your original door still only carries 1-in-3.
The catch
Switching is the better bet but not a guarantee. One time in three your first pick was already the prize, and switching walks you right off it — so staying still wins sometimes. Switching also only beats staying because the host knows the layout and always opens a losing door on purpose. If the host opened a door at random and might reveal the prize, the leftover game really would be 50/50 and switching would lose its edge.
Questions kids ask
If two doors are left, isn't it just 50/50?
No. The two closed doors are not equal, because the host did not open randomly — he knew where the prize was and made sure to open a losing door. That leaves your first pick at 1-in-3 and pushes the other 2-in-3 onto the single door you can switch to.
Does staying ever win?
Yes — staying wins whenever your very first guess happened to be right, which is about 1 time in 3. Switching just wins more often, about 2 times in 3, because your first guess is wrong twice as often as it is right.
What if the host opened a door without knowing where the prize was?
Then the trick disappears. A host opening a random door might accidentally show the prize, and in the rounds where he happens to reveal a goat, the two closed doors really are 50/50. Switching only wins 2-in-3 because the host knows and always avoids the prize on purpose.
Why does switching feel so wrong?
Because our brains treat the two leftover doors as a fresh, even choice and forget that the host's hint was loaded. Running the game hundreds of times shows the truth: the switch line settles near 2-in-3 and the stay line near 1-in-3, every time.
Talk about it
- Two doors are left and one prize — guess first: are they really equal, or is one a better bet?
- Why might a hint from someone who knows the answer be worth more than a hint from someone guessing?
- If you had to bet your dessert, would you stay or switch — and what makes you sure?
For grown-ups
This is the Monty Hall problem. The first pick wins with probability 1/3, so the prize is elsewhere with probability 2/3. Because the host knows the layout and always opens a losing door, no probability transfers back to the original pick; the entire 2/3 collapses onto the one remaining unopened door. Switching therefore wins 2/3 of the time and staying 1/3. The crux is the host's constraint — a host who opened a door uniformly at random, sometimes revealing the car, would leave a genuine 50/50 between the two closed doors.
Keep going
What else makes you wonder?
- What if there were 100 doors and the host opened 98 losing ones — would switching feel more obvious?
- Where else does a hint quietly change the odds without changing the thing you picked?
- How many rounds do you have to play before the win-rate stops jumping around and settles?