In §4.4 we are introduced to the concept of conditional probability. The notatio
ID: 1212395 • Letter: I
Question
In §4.4 we are introduced to the concept of conditional probability. The notation P(A | B) denotes the probability of event A occurring given that we know event B has occurred. Now consider the Monty Hall Problem introduced in the following video: https://m.youtube.com/watch?v=4Lb-6rxZxx0
•After watching this video, we know that if we are given the option to switch doors that, probabilistically speaking, it is in our best interest to switch.
•Explain why this is the case using your knowledge of conditional probability.
Explanation / Answer
In the Monty Halls problem, the players at the game show is given the choices of three doors, where behind two doors there are donkeys and behind one door there is a car. The strategy to win the game is that whenever the player is given a chance to switch, he should take it. The strategy can be explained by the help of conditional probability.
Assuming that you have picked door No.1, there are 3 cases:
Case
Door 1
Door 2
Door 3
Action of the host
1
Car
Donkey
Donkey
Door 3 or door 2
2
Donkey
Car
Donkey
Door 3
3
Donkey
Donkey
Car
Door 2
Now if we assume to follow case 1 – the probability of opening door 3 is 1/3
P(case 1 opening door 3) = P(case1). P(case 1/ opening door 3) = 1/3 *1.2 = 1/6
Adding the probabilities of the both ways that the doors can be opened becomes 1/3 +1/6
= ½
Now P (case 1 / open door 3) becomes = (1/6)/(1/2) = 1/3
If the players first choice was goat, then he can win 1 – 1/3 times by switching
That is 2/3 times.
Case
Door 1
Door 2
Door 3
Action of the host
1
Car
Donkey
Donkey
Door 3 or door 2
2
Donkey
Car
Donkey
Door 3
3
Donkey
Donkey
Car
Door 2