Monty Hall and the Let's Make a Deal Puzzle

The Situation:

• You are going to appear on Monty Hall's game show, Let's Make a Deal and fortunately, you already know how the game will be played.
• Monty, who knows all, will show you three doors.
• He will tell you that a billion dollars is hidden behind one door.
• It doesn't matter what's behind the other doors.
• You will choose the door that you think the money is behind.
• What is the probability of each door being the winning door?
• Monty will open another door revealing . . . Burma Shave.
• Monty will give you a chance to change your mind, to choose the other door.
• Do you keep the door you chose first, or do you switch to the other door?
• What is the probability of each door being the winning door now?
• Think about it, then scroll down below the Amazon.com advertisement to see how I would answer this question.

Here's how I replied who asked me that question:

Let me ask you a similar question. Monty Hall, the host of Let's Make a Deal, shows you one hundred doors. Behind one door is a billion dollars. The other 99 doors will only get you a prize worth less than one dollar. He lets you choose one door. You pick a door.

What are the odds that your door has the jackpot behind it? The odds are one in a hundred (1%; one chance in one hundred possible outcomes, or 0.01).

What are the odds that the jackpot is behind one of the OTHER doors? The odds are ninety-nine in a hundred (99%, 99 chances out of 100 outcomes, or 0.99).

I think we can safely agree

Now, you must remember, Monty Hall knows where the billion dollars is hidden. Knowing this, he opens 98 of the other doors. Every one of those doors has an inflated toy balloon behind it.

Only two doors are left, the door he didn't open, and the door you chose.

What are the odds that the jackpot is behind the door you picked?

What are the odds that the jackpot is behind the other door?

You know that one of the two doors that are left has a billion dollars behind it, and that the other has a penny behind it.

What is your gut feeling. Should you stay with door number one, the door that you picked, or should you change to the door that you didn't pick, the one that used to have a 99% probability of being correct when it was lumped with all the other 98 doors that are now open?

I bet that your "gut feeling" is correct.

Nothing has changed the probabilities. The probability that the billion dollars is behind YOUR door is still 1%. The probability that the billion dollars is behind one of the other 99 doors is still 99%. The fact that you know 98 of them are penny prizes doesn't change things. Some people will tell you that opening those 98 doors changes the probability on the other two doors to 50/50. Don't believe them.

In essence, Monty is asking whether you want to pick all 99 other doors, 98 of which you know are Burma Shave or ?????.

If you still aren't sure that I'm right, then let's try a smaller but still obvious test that we can perform in real life. Get out a deck of cards, and recruit two friends to help you.

With you, that makes three people.

1. Monty Hall
2. Sarah, who always switches, and
3. Keven, who always keeps.

Out of your deck of cards, pick out nine low cards and a King. Keep a pad of paper handy to keep score.

Step One:

• Monty shuffles, looks at the cards, then deals the cards face down.
• Remember, Monty knows where the King is.

Step Two:

• Sarah and Keven pick one card by placing their tokens on the card of their choice.

Step Three:

• Monty turns eight of the cards face up, showing that they are all low cards.

Step Four:

• Sarah switches her guess by moving her token to the other face down card

Step Five:

• Monty turns up both cards to show who would have won.

Step Six:

• Add one to the score of whoever had the winning card.

Now, do the whole thing again, starting again with step one. Do this ten times, and see how often Karl gets the King versus Sarah always getting the King. This should lead you toward finding the answer for yourself.

You can also do it with your original problem, by using three cards instead of ten.

About the Author

You can find some of his puzzles on his web page at <www.MAZES.com>. John has written materials that can help people get ready for the Driver's License exam. You can find these materials at www.DriverExam.org.

John is a computer programmer, consultant, web page designer, teacher, tutor and puzzle creator. John telecommutes throughout the world and is available to work for you. Please e-mail (webmaster (at) mazes.com).

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Copyright Information

If you use information from a source like this in a report, you should give proper credit, citing the author as a source. In other words, you must acknowledge where you found the information and who supplied it. This is a matter of professional courtesy. Here is an example of a footnote:

Footnote Example:
* John Knoderer, (webmaster (at) mazes.com), "American Numbering System and Place Values," http://www.mazes.com/AmericanNumberingSystem.html, September 10, 1997; revised May 21, 1999.

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