Wednesday, April 15, 2015

M is for . . .

The Monty Hall Problem

Image source: Wikipedia Commons

This math problem is dear to my heart because I distinctly remember where I was and what I was doing when the solution came: driving up one of those tight, cloverleaf, exit ramps off a highway. I couldn't wait to get home to write it down.

This problem gets its name from the original host of the TV game show, Let's Make a Deal, where a player faces three doors, two of which hide booby prizes and one hides something worth winning. Here 'tis:

"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 No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?" (Wikipedia)

What do you think? You've picked a door. After that, the host shows you that one of the two doors you didn't pick hides a booby prize. You could stick with your first choice or switch. Seems a fifty-fifty chance now, doesn't it? But that's where the math comes in.

The answer is that you should always switch. In fact, you have a 2/3 chance of winning the car if you do switch and only a 1/3 chance of winning if you don't. Surprised?

Here's the key: At the beginning, when you selected one door out of the three, you probably picked the wrong one. Why? With 2 goats and 1 car, random picking means there's a 2/3 chance you picked a goat. Therefore, odds are the car will be yours if you switch.

Cool, huh?

Do you (or did you ever) watch game shows? Would you prefer a goat over a car?


This week's short story giveaway, The President and the Pea, features a sci-fi tale inspired by The Manchurian Candidate.


  1. I probably wouldn't switch. It's when I start second guessing myself that I mess up.

  2. I am surprised. Teachers always said to stick with your initial gut instinct and most often when a student changed their answer at the last minute they were wrong and what they first thought was right.

  3. That's interesting to know. I would probably switch just because I tend to second guess myself anyway.

  4. So that's where the concept behind Let's Make A Deal comes from. :)

  5. I would not switch. I was never good at math anyway. Gut instinct is more my speed. Then, of course, if I wound up with the goat, I would have been kicking myself and saying “Darn it!” “I knew I shoulda switched!”

  6. ha. Before reading this I probably would have stuck with my first choice. Thanks for the insight!

  7. I am not good with games of probability, and I guess following the advice of this game I could get better. However, if I were on a show or put on the spot to make a decision between one of three doors, oh boy would I feel indecisive!

  8. Now that's interesting. Need to remember to switch. I think my instinct would have been to stay with my first choice.
    Inventions by Women A-Z

  9. Whatever I did, I would probably end up with a goat!

  10. I literally had to read the problem - and the solution - twice before I got it :)

    Such a silly goat... cheesy, yes, I know :)

  11. Maybe I wanted the goat... ;)
    But seriously, it's a fascinating maths problem.

  12. This sounds fascinating. Math always confuses me! :)

  13. I totally watched the games, but here's another conundrum to add to the statistics of the choice: If you change your vote, only half of the time will that be the right choice.

    Even though, yes, the odds are that you had a wrong door going into the challenge, if you do a statistical regression for after the fact choices, it's a straight fifty fifty because the probability of it being one of the two remaining doors gets upgraded. That is to say, if you swap doors, you're right half of the time and wrong half the time (CRAZY).

    The mind boggling part about this conundrum is that if you pulled out a coin and flipped it to check which door to pick, even if the coin landed on the door you already had picked, the 33.333% chance door, the coin is going to be correct exactly half of the time.

    I love math, but I am no mathemagician

    1. For more mathematical conundrums, look up the math monsters. They will break your mind because they both obey and break logic: SO. MUCH. FUN. (Nova has a good explanation on the Fractal episode)

  14. I wish I would've known this trick years ago, so I could've stuffed my purse with everything under the kitchen sink, and headed straight to the show! I'll have to see if I can stump my son with this math problem. Thanks Tamara!


  15. Mmmm! Are you sure it works? Interesting.

  16. I love reading about stuff like this. But wait! The wikipedia version is confusing: "You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat."
    They make it sound as though the contestant chose door no. 1 and the host deliberately opened door no. 3!

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