“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?

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

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.

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.