Doubt

Question

Suppose we are playing a game.You have to choose 1 out 3 doors.Behind 1 lucky door ,we have 1 million dollars.You picked 1 st door.But i opened the 3rd door and found nothing.Now,if i give you another chance ,will you change your choice?                          (Use probabiliy estimates to answer)

Comments

Asked by kevin spacey in the movie 21 :)

Solution of the problem is given below in the answer section.

Answer 1

Suppose there are three doors

Door 1 Door 2 and Door 3

Suppose our strategy is when the referee opens a door for the second time we will not switch our first choice

Let us assume that million dollar prize is behind the door 2

Now, you will have three cases

You choose door 1 and the referee opens door 3

Since you are not switching you will lose.

You choose door 2 and the referee opens door 1 or door 3

In this case you win.

You choose door 3 and referee opens door 1

In this case you lose

So, your chance of winning is (1/3) with the strategy of not switching.

Now, suppose your strategy is that you will switch the door when the refree opens the door for the second time.

This time also you will have three cases

Case 1 You choose first door referee opens third door

Since you are switching you win.

Case 2 You choose second door referee opens door 1 or door 3

In this case you will lose

Case 3 You choose third door referee opens door 1

In this you will win since you are switching

So, the chance of winning when you are switching is 2/3.

So, if I have another choice I will definitely switch.

Comments

why?

if referee choose door2, then I will loose tat case too

but why not he choose it?

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referee knows that there is prize behind door 2.

Now, if he shows that to me then the game is over but here he wants to give me another chance.

Thats why he will not choose door 2.

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ok..

Answer 2

Its a very famous problem in probability called The Monty Hall problem.

Lets make the game a bit interesting by having a million gates, now you choose one gate. The probability of you being correct is $\frac{1}{million}$ . Now I open all gates which are empty except one. Now will you choose to switch?

Obviously you will switch because you knew from the start that you were wrong with a tiny probability to win. In this case its clear switching is same as winning.

Now coming to your question initially you chose a door, the probability of you being right is $\frac{1}{3}$. And the probability of having the lottery in behind one of the two is $\frac{2}{3}$ with each gate having $\frac{1}{3}$ probability of having prize money(i.e probability $\frac{2}{3}$ is equally distributed amongst the unchosen gates) . $\because$ The total probability must be 1, once I open a gate which is empty the probability of you being correct is still $\frac{1}{3}$ but the probability is redistributed among other unopened gates(in case the only unopened gate ) , $\therefore$ the unopened gate gets probability of $\frac{2}{3}$.

Hence you must switch to have a better chance of winnig. Although you may loose if you chose the correct gate initially, but chances of that are $\frac{1}{3}$ < $\frac{2}{3}$ which you get after switching.