Question

Question 1:

Consider the following Monty Hall problem. Suppose you are on a game show, and you are given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what is behind the doors, opens another door, say #3, which has a goat. Here we assume that the host cannot open the door to expose the car and when he can open either of two doors, he chooses between them at random. What is the probability of winning the car if you switch your choice and pick door #2?

(1) 1/3

(2) 1/2

(3) 2/3

(4) 3/4

Question 2:

Consider the Monty Hall problem described in Question 1 again. Now we modify it so that you are given the choice of four doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what is behind the doors, opens two other doors, say #3 and #4, each of which has a goat. Here we assume that the host cannot open the door to expose the car and when he can open two out of three doors, he chooses them at random. What is the probability of winning the car if you switch your choice and pick door #2?

(1) 1/4

(2) 1/3

(3) 1/2

(4) 3/4

Question 3:

Now we consider a variant of the Monty Hall Problem described in Question 1. In this variant, you have selected one of the three doors, say Door #1, the game show host slips on a banana peel and accidentally pushes open another door, say Door #3, which just happens not to contain the car. What is the probability that you will win the car if you stick with your original selection, i.e., Door #1.

(1) 1/3

(2) 1/2

(3) 2/3

(4) 3/4

Question 4:

Either Alice or Betty is equally likely to be in the shower. Then, you hear the person in the shower singing. You know that Alice always sings in the shower, while Betty only sings 1/4 of the time. What is the probability that Alice is in the shower.

(1) 5/7

(2) 2/3

(3) 3/4

(4) 4/5

Question 5:

Consider the example described in the on-demand lecture on Simpson’s Paradox. How can we make sure that Simpson’s Paradox cannot occur in an experiment like the example given in the on-demand lecture. The following choices assume the setting described in the example given in the on-demand lecture.

(1) The relative proportions of men and women among the subjects who receive the new treatment are the same, or approximately the same, as the relative proportion of men and women among the subjects who receive the standard treatment.

(2) The number of men who receive the new treatment and that of women are the same. Also, the number of men who receive the standard treatment and that of women are the same.

(3) The number of men who receive the new treatment and that of women who receive the standard treatment are the same. Also, the number of men who receive the standard treat and that of women who receive the new treatment are the same.

(4) It is a matter of chance. There is no way to avoid Simpson’s Paradox.

Answer 1

solution :- as Let us reperesent u doors car goat 2 I goat door 3 door 1 door a Condition - 1 a you choose doon 1 Host opens because both contain goats. any of remaining doors - If you switch choice your ? you get Goat !