Let's play a game ❤️
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 3, and the host, who knows what is behind the doors, opens another door, say No. 1, which has a goat. He then says to you, "Do you want to pick door No. 2?"
Is it to your advantage to switch your choice?
Answers
A very interesting and good question indeed. That's something related to the concept of probability.
Let us assume that the three doors are named door 1, door 2, and door 3 respectively. Door 1 has a car behind it, Door 2 and 3 have goats behind them. The rule is, if I'm not wrong, that the contestant has to choose any one door out of three doors and after he chooses, the host will reveal door no. 2 or 3 and then will ask him that he wants to switch or not and the contestant needs to answer. We need to find that the constestant has more chances of winning the car if he switches the door or not. Let's Do so.
Case#1: Let the contestant choose door number 1. Here, the host opens door 2 or 3. If the contestant switches, he will not win the car. If he doesn't switch, he wins the car.
Case#2: Let the contestant choose door 2 and the host reveals door 3. If he switches, he wins or else, he looses.
Case#3: The contestant chooses door 3 and the host opens door 2. Then, if he switches, he wins or else, he looses.
Let's tabulate the above results. Now, we have the following data:
____________________
Door chosen = 1
Switches: Looses
Doesn't Switch: Wins
____________________
Door chosen = 2
Switches: Wins
Doesn't Switch: Looses
____________________
Door chosen = 3
Switches: Wins
Doesn't Switch: Looses
____________________
From the above data, It is clear that he wins 2 out of every 3 times he switches and looses one out of every three times. Also, If he doesn't switch, he looses 2 out of 3 times and his chances of winning are only one out of three times.
In other words, Probability of wining after switching = 2/3.
So, it is clear that he should switch if he want to win as after switching, the chances of his winning are more than the chances of his winning after not switching.
So, the contestant should switch the door when prompted after the opening of the door which has a goat by the host.
I am given the choice of three doors .
Behind one door , there is a car .
Behind the other doors , there are goats .
I pick a door and the host opens another door , which has a goat .
We are given a situation where we have to either switch or choose door 2 so that we get a high percentage of getting a car . Off course no one would want a goat .
If we look into the problem deeply we will find three outcomes that may arise :-
Let us keep the car in door 2 for trial and error .
Possibility 1
The door 1 is chosen . Then the host will open door 2 or he may open door 3 . If the user switches then he does not win the car .
Possibility 2
The door 2 is chosen . Then the host will open either door 1 or door 3 . Here if he switches to door 2 , he will win the car .
Possibility 3
The door 3 is chosen . The host will open door 1 and if the user switches now he will win otherwise he will lose .
Out of 3 possibilities , 2 are favourable and hence upon switching the probability becomes 2/3 .
If we do not switch probability becomes 1/3 .
Hence switching is better .