in monty hall problem instead of 3 doors there are 20 doors and behind 19 there are goats and behind 20th is a shiny new car.Host open 18 doors and two doors are close and you have to switch or stick to your chosen door.If you do not switch,you have the expected 1/19 chance of winning car however. what is the probability of you now switch doors?
To solve the Monty Hall problem with 20 doors, we can adapt the same logic used in the original problem with 3 doors. Let's break down the steps and calculate the probability.
1. Initially, when you select a door, the probability of choosing the one with the car is 1/20. Therefore, the probability of choosing a door with a goat is 19/20.
2. When the host opens 18 doors with goats, the original probability of the door you selected containing a car remains 1/20. However, the probability that you chose a door with a goat has now increased to 19/20 since the host has revealed the location of the goats.
3. Since the host offers you the choice to switch doors, you need to consider the probabilities based on your decision.
a. If you stick to your original door, the probability of winning the car remains 1/20. This is because the host opening the other 18 doors does not affect the probability of your initial choice being correct.
b. If you switch doors, the probability of winning the car is now 19/20. This is because when you switch, you effectively choose one door out of the remaining 19 doors, knowing that the probability is concentrated in those unopened doors.
Therefore, if you decide to switch doors, the probability of winning the car becomes 19/20.