Visualizing the Solution to the Monty Hall Problem

The “Monty Hall Problem” (aka “Game Show Problem”) was made famous by The World’s Smartest Woman, Marilyn Vos Savant, in her Parade Magazine Q&A column.

Here’s the original question, followed by Vos Savant’s answer:

Suppose you’re on a game show, and you’re given the 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 says to you, “Do you want to pick door #2?” Is it to your advantage to switch your choice of doors?
Craig F. Whitaker
Columbia, Maryland
Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?

It’s famous because a lot of smart people, including Math professors at universities, criticized her answer as WRONG, when she was in fact RIGHT. Most people think the odds are still 1 out of 3. But the answer is indeed 2/3.

Part of the problem is that Marilyn’s answer isn’t very intuitive, nor are the solutions given on the internet (see Wikipedia’s explanation, for example).

So, here’s my stab at trying to make the solution visual by adding up the possibilities (bottom-layer or “leaf nodes”) in a decision tree.

Start

Monty Hall (emcee for the TV show “Let’s Make a Deal”) gives you the choice of 3 doors. Under one of the doors is a prize. Let’s name the doors: “WIN”, “FAIL”, and “fail”.

Chance of Winning: 1/3
Chance of Losing: 1/3+1/3 = 2/3

Scenario 1: Monty Hall gives you the option of switching but doesn’t show you an empty door

There are six possibilities: WIN→FAIL, WIN→fail, FAIL→WIN, FAIL→fail, fail→WIN, fail→FAIL; each with a 1/6 probability (see diagram).

Chance of Winning: 1/6+1/6 = 2/6 = 1/3
Chance of Losing: 1/6+1/6+1/6+1/6 = 4/6 = 2/3

Recommendation: Your chances of winning haven’t changed, so don’t switch.

Scenario 2: Monty Hall shows you an empty door (equivalent to removing a fail — compare this with previous diagram)

There are only 3 possibilities, WIN→FAIL, FAIL→WIN, fail→WIN; each with a 1/3 possibility (see diagram).

Chance of Winning: 1/3 + 1/3 = 2/3
Chance of Losing: 1/3

Recommendation: Your chances of winning have improved, so ALWAYS switch

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