Question

Quite a few years ago ,a popular show called lets make a… quite a few years ago ,a popular show called lets make a deal appeared on network television.contestants were selected from the audience . each contestant would bring some silly item that he or she would trade for cash prize or prize behind one of three doors. suppose you have been selected as a contestant on the show.you are given a choice of three doors. behind one door is a new sports car. behind the other doors are a pig and a chicken -booby prizes to be sure.let's suppose you choose door no one.before opening the door the host who knows what is behind each door ,opens door two to show you chicken.he then asks you ,"would you be willing to trade door one for door three" what should you do ? REQUIRED TASKS 1,Given that there are three doors ,one of which hides a sports car ,calculate the probability that your initial choice is the door that hides the sports car .what is the probability that you have not selected the correct door? 2.Given that the host knows where the sports car is , and has opened door 2, which revealed a booby prize does this affect the probability that your initial choice is the correct one? 3.Given that there are now only two doors remaining and that the sports car is behind one of them,is it your advantage to switch your choice to door 3?( Hint: Eliminate door 2 from consideration. The probability that door 1 is the correct door has not changed from your initial choice. calculate the probability that the prize must be behind door 3. This problem was discussed in Movie 21 starring Jim sturgess, Kate Bosworth and Kevin Spacey)

Answer 1

using a tree diagram I have explained the answer.

given,

door 1 = sports car

door 2 = pig and chicken

door 3 = booby prizes

I have used the name Tony for example.

One solves this problem by comparing the probability of choosing the car if you stick with your original choice to the probability of choosing the car if you switch after Tony opens the one door. Note that the car has an equal probability of 1/3 of being behind Door 1, Door 2, or Door 3.

First, suppose that your strategy is to stick with your original choice of Door 1. Then you only win if the car is behind Door 1 so that your probability of winning is 1/3.

Next, suppose that your strategy is to switch doors. We break this into three cases:

If the car is behind Door 1, Tony will open either Door 2 or Door 3 to reveal a pig & chicken or booby prizes. You switch to the other of Door 2 or Door 3, and in either case, you switched to a door with pig & chicken or booby prizes behind it (remember, the car is behind Door 1).

If the car is behind Door 2, Tony will open Door 3. This is because he always opens a door with pig & chicken or booby prizes behind it, and he can't open Door 1 because that was your original choice. So the only door you can switch to is Door 2, which is the door with the car behind it. Ding! You win!

If the car is behind Door 3, Tony will open Door 2. This is because he always opens a door with pig & chicken or booby prizes behind it, and he can't open Door 1 because that was your original choice. So the only door you can switch to is Door 3, which again is the door with the car behind it. Ding! You win!
So if your strategy is to switch doors, you win 2/3 = 1/3 + 1/3 of the time. (Remember, the probability is 1/3 that the car is behind any particular door.) Therefore, a better strategy is to switch doors - the calculated probabilities indicate that you are twice as likely to win if you do this!