Microsoft

## Interview Question

Program Manager Interview

-Redmond, WA

# You are on a game show. There are three doors, behind one of which is a prize and the other two is a chunk of coal, and the host knows which door holds the prize. You choose door #1. Before it is opened, the host opens door #3 and reveals a lump of coal. You have the choice to stick with the door you chose originally or switch to door #2. What do you do?

Tags:brain teaser, puzzle

10

Interview Candidate and Anonymous are right. This is also known as the Monte Hall problem

Tony on

10

Your choice splits the doors in two sets. Set A contains the door you selected, and the probability that is a prize behind this door is 1/3. The set B contains all remaining doors, and the probability that the winning door is somewhere in there is 2/3. By removing one door, which all have the success probability of zero because there's coal behind them, from set B, only one door remains in B, but the overall probability for success in set B is still 2/3. Therefore you must switch.

Manish on

10

Interview candidate is right. You got 1/3 chance that prize is behind door #1 and you lose if you switch. And you got 2/3 chance that prize is behind either door #2 or #3. Since the host will always eliminate the wrong one. 2/3 chance will be allocated on the left one.

Anonymous on

2

Averee has given the best explanation. Before anyone writes any further comment or contradicts what he says, do yourself a favor, go on youtube and watch the video on monty hall problem - you will not object to averee's solution after that.

Akhil on

2

Switching is the correct answer.

listen on

1

PS. Watch it twice.

John G. on

1

You must switch the door. Imagine the same situation with 100 doors and you will understand.

Amit Garg on

1

Thank you Akhil. I'd like to add that the youtube video "Monty Hall Problem for Dummies - Numberphile" explained it very well.

John G. on

2

What everyone who is answering "stick to original door" is missing the host's knowledge of which door has the prize and his actions after the first door is chosen. Host's action puts the odds of prize behind the switched door as 0.5. However, the original door still has a 0.33 probability of the prize. Yes I know it is a difficult concept to grasp - so you all need to watch the monty hall problem on youtube.

Akhil on

4

Really??? I can see why they ask the question if so many people have so little understanding of probability. There are NOW two doors to choose from and there is one prize. DAAAAA It doesn't matter how many there were to begin with or which one of the two you originally chose. There are now two doors and each one has an equal chance of containing the prize so your odds of success do not change at all regardless if you stay with your original choice of switch to the other. The odds of your original door being the correct one were 1/3 when there were 3 option but the odds of your door being the correct one CHANGE to 1/2 when you eliminate one of the options. To say the probability of your door being correct are still 1/3 is to say that your odds of success are still one third if you were to pick door 3 even though you know it does not have the prize. As an interviewer I could have easily eliminated the majority of the applicants above.

whack on

1

What about sticking to your original decision and not deviating without knowing the facts(Data). i will stick with my decision and might fail, sure but at list i will not make decision on just the 1/3 and 1/2 number game.

Ashish on

0

Tarik, You are correct that the right choice has increased from 1/3 to 1/2, but you have to account for variable change in this instance. Think of the same problem, but with different numbers. 100 doors, same conditions. You pick 1 door, with a 1/100 chance of being correct. The game show opens 98 doors with coal (because he knows where the prize is). He now offers you the chance to keep your door or switch to the last remaining door. In this Because of variable change you will always be better off switching doors.

Averee on

0

Scratch that, the right choice hadn't increased from 1/3 to 1/2. It's late and I have no idea why I typed that.

Averee on

6

I can't believe that some of the answers up here actually state that you should switch. -------------------------- Before the host showed you that door#3 was hiding coal, your chances to pick the right door where 1/3. Now that you know that coal is behind door#3, you only know that 1 of the 2 remaining doors leads to coal while the other leads to the prize. While your probability of making the right choice has increased from 1/3 to 1/2, this probability still applies to both of the doors. You can't differentiate them. Seriously.

Tarik on

25

Switch doors. When you chose door #1, there was a 66% chance that the prize was not behind that door. When the host revealed the coal, there was still a 66% chance the prize was not behind the door you chose. Thus, you have double the odds of getting the prize by switching to door #2. The key to this puzzle is that the host knew which door has the prize.

Anonymous on

13

That is not true. The probability went from 1/3 to 1/2 once the number of doors reduced to 2. However, statistically speaking, your chances of finding the treasure are now even. So it should not matter which door you pick.

Murthy on

3

Everyone is wrong. You stay with your first pick, the odds becoming slanted greatly against the other door.

scooter58 on