3 Doors and Angel

Problem :

You are given a choice of three doors by an Angel. You can choose only one of the doors among the three. Out of these three doors two contains nothing and one has a jackpot.

After you choose one of the doors angel  reveals one of the other two doors behind which there is a nothing. Angel gives you an opportunity to change the door or you can stick with your chosen door.

You don’t know behind which door we have nothing. Should you switch or it doesn’t matter?

Soltuion:

You choose one of the door. So probability of getting the jackpot – 1/3.

Let’s say that the jackpot is in Door no 1 and you choose Door no 1. So the angel will either open door no 2 or door no 3. Let’s look at the sample space of this Puzzle.

Case  ->             Door1                                   Door2                                Door3
Case 1 :            Jackpot                                  Nothing                               Nothing
Case 2 :            Nothing                                  Jackpot                               Nothing
Case 3 :            Nothing                                  Nothing                              Jackpot

Want to keep your guess:

Let’s suppose that you guessed correctly. Then it makes no difference what the game show host does, the other door is always the wrong door. So in that case, by keeping your choice, the probability that you win is 1/3 x 1 = 1/3.

But let’s suppose you guessed incorrectly. In that case, the remaining door is guaranteed to be the correct door. Thus, by keeping your choice, the probability of winning is 2/3 x 0 = 0.

Your total chances of winning by keeping your guess is: 1/3 + 0 = 1/3.

Want to change your guess:

Again, let’s suppose that you guessed correctly. By changing your guess the probability that you win is 1/3 x 0 = 0.

But let’s suppose you guessed incorrectly. Again, this means that the remaining door must be the correct one. Therefore by changing your choice, the probability of winning is 2/3 x 1 = 2/3.

Your total chances of winning by changing your guess is: 2/3 + 0 = 2/3.

Hence it is advisable to switch.

One Thought on “3 Doors and Angel

  1. This in an old question whose answer has been tricked on through generations, even the most educated ones take this explanation to be true. But this includes a flaw. Let me try to show you another view point before I cite the flaw:
    lets say the prize is actually behind door 2. If we choose door 1, they open 3. If we choose 3 they open 1. Had we chosen door 2 they still had 2 options to open- door 1 and door 3.
    Our guess in the first two scenarios is incorrect whereas in the later two it is correct. Whether we change it or not, the probability remains 2/4, i.e. 1/2; it is neither 1/3 nor 2/3.
    The problem is that they don’t consider the fact that hosts open one wrong door always. So there remains only 2 doors. One has the prize and the other doesn’t, as simple as that.
    Shown one door containing nothing, there are 2 doors with equal probability of having the prize. It’s just trick to keep you thinking.
    Remember the dummy coins we added while solving Probability exercises for IIT-JEE. The extra door here is the dummy coin.

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