Test It With Cards

You can settle this at your kitchen table. Take three playing cards: one red for the car, two black for goats. Shuffle face-down and lay them out.

Pick one without looking. Now play the host: peek at the other two and flip a black card. Switch to the last face-down card, note the result, and repeat. Do many rounds always staying, then many always switching.

The pattern shows up fast — switching wins clearly more often, roughly twice as much. Doing it by hand convinces you in a way no argument can.