There are three boxes that contain numbered tickets. The following table indicates which tickets are in each

box.

Box Set of Tickets in the Box

1 {1,1,2,2,2,3}

2 {1,1,1,2,2,3}

3 {1,2,2,3,3,3}

Answers to the questions below should be given as fractions.

a) A ticket is randomly selected from Box 1. Let the random variable, X, represent the number selected from the box. Give the probability distribution of X.

b) Suppose two tickets are drawn from Box 1 with replacement (after a ticket is drawn it is put back in the box). Let the random variable X represent the sum of the two tickets drawn from the box.

(i) What is the sample space of X?

(ii) What is P(X > 2)?

(iii) What is P(X = 4)?

(iv) What is P(X = 4 | First ticket drawn is a 1)?

c) Suppose one of the three boxes is randomly selected and a ticket is randomly drawn from the selected

box. Let X represent the number drawn from the box.

(i) What is P(X=1|Box 1 is selected)? P(X=1|Box 2 is selected)? P(X=1 | Box 3 is selected)?

(ii) What is P(X=1)?

(iii) What is P(Box 3 was selected | X=2)?