Question

“Let’s Make a Deal.”Marilyn vos Savant, who is listedin Guinness Book of World Records Hall of Fame for“Highest IQ,” writes a weekly column in the Sunday newspaper supplement Parade Magazine. Her column, “AskMarilyn,” is devoted to games of skill, puzzles, and mind-bendingriddles. In one issue (Parade Magazine, February 24, 1991), vos Savant posed the following question:

Suppose you’re on a game show, and you’re given a choice of three doors. Behind one door is a car; behind the others, goats. You pick a door—say, #1—and the host, who knows what’s behind the doors, opens another door—say #3—which has a goat. He then says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice?

Marilyn’s answer: “Yes, you should switch. The first door has a 13 chance of winning [the car], but the second has a 23 chance [of winning the car].” Predictably, vos Savant’s surprising answer elicited thousands of criticalletters, many of them from PhD mathematicians, who disagreed with her. Who is correct, the PhDs or Marilyn?

Short answer

The result is that Marilyn is right.

Step-by-step solution

Important formula

The formula for probability is$\mathbf{P}\mathbf{=}\frac{\mathrm{favourable}\mathrm{outcomes}}{\mathrm{total}\mathrm{outcomes}}$

Decide who is correct.

Let’s assume that G= goat and C =car.

Here the three options are to arrange a car and two goatsbehindthe three doors are CGG, GCG, and GGC.

Now consider the first option CGG.

If I choose the door with number 1, then the host can open the door with number 2 or 3 because behind both doors are goats.

In both cases it doesn’t matter if the host opens the door with number 2 or 3, if I switch our choice, then I win a goat.

Also in both cases, it doesn’t matter if the host opens the door with number 2 or 3, if I don’t switch our choice, then I win a car.

Choose the next option GCG

If I choose the door with number 1, then the host can open the door with number 3.

If I switch our choice, then I win a car.

If I don’t switch our choice, then I win a goat.

Now, choose the next option GGC.

If I choose the door with number 1, then the host can open the door with number 2.

If I switch our choice, then I win a car.

If I don’t switch our choice, then I win a goat.

This shows that if I switch our choice then I win a car in 2 or 3 cases which means I have a$\frac{2}{3}$ chance of winning the car.

Andif don’t I switch our choice then I wina car in 1 or 3 cases which means I have a$\frac{1}{3}$ chance of winning the car.

Therefore, the result is that Marilyn is right.