Friday, September 09, 2005

To Switch or not to Switch?

Let's say you are on a gameshow. The hosts shows you 3 doors, 2 have goats behind them while 1 has a car. The host then asks you to pick one door. After you do that, he opens one of the doors that you did NOT pick and reveals a goat. He then asks you if you would like to change your choice and switch to the other door, or keep your original choice.

The question: Do the odds of you getting the car increase if you switch? Or is it 50%, so it doesn't matter if you switch or not?

The puzzle is actually a very famous one. In 1990, Marilyn vos Savant published her answer to the puzzle in her newspaper columm and immediately got thousands of letters claiming she was wrong, some of which were PHD holders and professional mathematicians.

Anyway I got this puzzle from a book I am reading now called "the curious incident of the dog in the night-time" by mark haddon, a story about an autistic teenager with amazing memory and logic skills who tries to solve the murder of his neighbour's dog in the style of his favourite detective, Sherlock Holmes. It's quite an amazing read so far, highly recommended.

Update: The answer (this is for you, Frankie):

1st Choice
/ | \
G1 G2 C
S/ \NS S/ \NS S/ \NS <--- 2nd choice after Host reveals one goat
C G1 C G2 GX C

G1 - Goat 1
G2 - Goat 2
GX - Either Goat
C - Car
S - Switch
NS - Not Switch

As you can see from all the possible outcomes in the badly illustrated diagram, you get 2 out of 3 chances to get a car when you switch, and only 1 out of 3 chances when you don't.


  1. i read the book too, intrigued by the title and cover. rather sad, the boy..

    i dun care much about probability, but since i always kick myself for switching (and getting it wrong) i now consciously tell myself to just stick to the original decision. you win some, you lose some..

  2. I thought you should switch? The first time u try u have a 33% chance. Now that one door has been ruled out, you have 50% chance of getting a car.

  3. that book is awesome! I remember that puzzle're supposed to choose the other door; right? I don't understand it, but supposedly it increases your chances, I think. Me no good at math so I don't understand it. :)

  4. Answer: Yes you should ALWAYS switch. If you switch your probability doubles from 33% to 66%. It's to do with conditional probability.

    Yes I found it a bit hard to comprehend at first, because I felt like it was a 50% chance (since there were 2 doors left and one has a goat while the other has a car). But in the book they actually did a diagram with all the possibilities, and it makes it much more clearer.

    Also pple have run computer simulations of the problem like millions of times , and the percentage of them getting the car is around 66% when they switch.

    It's an interesting case how our "common sense" can sometimes be so misleading.

  5. how come ar? i mean what if you're damn lucky that 1st door you choose is the car le? there's still a chance right?

    i've always wanted to read that book but too lazy...

  6. Zhi Yang : Ya of cos there is a chance if you just stick with your original chance, but it will be only 1/3 compared to switching which gives you 2/3.

    But there are websites that have programs that run a simulation of the program a large number of times, and you can clearly see that switching yields twice as many number of successes as when you don't.

  7. *laughs* I learnt that in high school. It traumatised me for an entire week. once you draw diagrams and do the math, it's actually right.

    never read that book, but it's on 3 for 2 at borders. I may consider it.

  8. Silvermyst: Ooooh a chick who digs math puzzles. You know, I had dreams about gals like ya. Very nice dreams. So when will you be coming back to Singapore? Maybe we could you know, do something sometime. ;)

  9. i am not convinced. it's definitely a 50-50 situation.

  10. I rather have a goat because now petrol price not cheap instead goat give me milk :)

  11. Anonymous: K think of it this way. Instead of 3 doors and 2 goats there were 99 goats and 100 doors. (still only 1 car)

    You pick one door. Your chance of getting the goat is 99% Chance of a car ? Only 1 %

    Host opens 98 doors and shows all goats, leaving one door unopen.

    Do you switch? Or stick to your choice and hope that you were really really lucky to pick the correct one out of a 100 in the first place?

    If you say stick, then what if it was a million doors. Will you still stick?

    Wai Hong: Ha, and can ride on the goat as well !

  12. I heard that book is awesome. I really want to read it.

    I can't wrap my little brain around this probability me, it still seems like you have a 50-50 chance of picking the right door now that one is eliminated, but the chance that you picked the right door to begin with is, how would switching to the other door increase your chances in reality?

    Draw the diagram, Bert!

  13. ha! C'mon, C'mon..Don't be scared of the sedition act and give me an update - yeah!

  14. I guess I was mistaken the first time. Guess this time, I'll still stick to the original decision. A goat isn't really such a bad thing anyway.