1. Monte Hall, twisted Consider the following twist on the monty hall problem (s

Question

1. Monte Hall, twisted Consider the following twist on the monty hall problem (see video on Rhea to recollect) There are 3 doors, behind one of which there is a car, and the remaining two have goats. You and a friend each pick a door. One of you will have definitely picked a goat. The host then opens one of the two chosen doors (i.e. yours or your friend's), shows a goat and kicks that person out (if both of you have chosen goats, he opens one door at random) The other person is given the option of staying put, or switching to the one remaining door (a) When originally picking your doors, should you choose your door first, or let your friend go first? Does it make a difference? (b) It turns out that the host has just eliminated your friend. What should you do? (i.e. stick with your original door, or switch?) What are the probabilities of winning in each case? Solution: (a) It makes no difference whether you or your friend chooses first; you have equal probability of choosing the curtain with the car (b) In this game, you should stay put because the chances are that the car is behind one of the two curtains picked by you and your friend. He's already gotten the goat, and so you get the car with probability

Explanation / Answer

The probability of choosing a door out of 3 doors is 1/3

There are 3 doors out of which only 1 door has the car

Probability of selecting the door with car =1/3

Probability of selecting a door with goat is 2/3 as there are 2 doors with goat

a)

Now since there is only 1 door with the car

In the game there is a equal chance of 1/3 that you or your friend will win the car

Hence it does not matter if you go first or he goes first as the probability will not change ie probability of winning will still remain the same= 1/3

b) Here one door is eliminated as your friend has got a goat

So you have the chances between 2 door to get the car out of 3 doors

Even if you change your option you still have a probability of 2/3

ie if there are doors

1 2 3

and your friend chose 3 and got a goat

you have the option of choosing between 1 and 2

probability=2/3 ( doors 1 and 2 out of 3 doors) to win a car

You already know that only 1 door has the car

So it is a 50-50 chance between door 1 and 2

so you should stay put as the probability of winning is still the same ie 2/3

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