The Monty Hall problem

1. Step 1 · Your first pick is the car

   $\displaystyle 1 \text{ of } 3 \text{ doors } = \frac{1}{3}$

   Your first pick is right 1 time in 3. Then the other closed door hides a goat, so switching loses.

2. Step 2 · Your first pick is a goat

   $\displaystyle 2 \text{ of } 3 \text{ doors } = \frac{2}{3}$

   Now the host must open the other goat door, since he never shows the car. The door left closed has to be the car.

3. Step 3 · Switching wins

   $\displaystyle \text{whenever your first pick was } a \text{ goat } = \frac{2}{3}$

   The host’s door tells you nothing about your own pick, which stays at 1/3. All the rest of the chance moves to the other door.

Answer$\displaystyle \frac{2}{3} \approx 66.7\%$