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?
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.