l u the Hall ne Solutiona You win if the chosen door has a c

Solution

(a) You win if the chosen door has a car behind it.

You choose a door (say 1 without loss of generality).

P(1stdoor has a car)=1/4

P(door 2,3, or 4 has a car)=3/4

Now, if among all three doors that you did not choose, 2 such doors are opened that do not have a car.

If you go stay with your choice i.e., door 1 then, P(you win)=P(1stdoor has a car)=1/4

If you switch to the only unopened door among 2,3 and 4

P(you win)=P(door 2,3 or 4 has a car)=3/4

Thus P(win)=0.25 if you do not switch, P(win)=0.75 if you switch.

(b)

In case of n doors :

P(chosen door has a car)=1/n

P(car is in any of the n-1 doors except the chosen one)= 1 - 1/n

Now, after opening n-2 doors, only 1 door is unopened among the rest of n-1 doors (those except the chosen one)

Now, if you do not switch P(win)=1/n

but, if you switch P(win)= 1 - 1/n = (n-1) / n

Note- the easier way to comprehend this is that interpret the implications of the following two decisions-

If you stay with your choice: you have access to only one door (that you choose)

If you switch: you have access to all other n-1 doors (except the one you choose earlier). If any of those doors has the car behind it, you win. n-2 of those doors have already been opened by the host and you are going to open the only remaining door of those n-1 doors. The host has opened only those doors (among n-1) which did not have a car behind them so, if any of these doors had a car then you are sure to win after switching. It is like you get access to all of n-1 doors that were remaining.