Question

Suppose that 2 balls are chosen without replacement from an urn containing 5 white and 8 red balls. Let X1 equal 1 if the 1st ball selected is red, and let it equal 0 otherwise. Let X2 equal 1 if the 2nd ball selected is red, and let it equal 0 otherwise. (i) Give the joint probability mass function of X1 and X2. (ii) Are X1 and X2 independent

Answer 1

The probability mass function :

P(X1 = 1, X2 = 1) = 5/13 * 4/12 = 5/39

P(X1 = 1, X2 = 0) = 5/13 * 8/12 = 10/39

P(X1 = 0, X2 = 1) = 8/13 * 5/12 = 10/39

P(X1 = 0, X2 = 0) = 8/13 * 7/12 = 14/39

Explanation:

Given :

Number of white balls = 5

Number of red balls = 8

Total number of balls = (5 + 8) = 13

P(X1 = 1, X2 = 1) = 5/13 * 4/12 = (5/13 * 1/3) = 5/39

P(X1 = 1, X2 = 0) = 5/13 * 8/12 = (5/13 * 2/3) =10/39

P(X1 = 0, X2 = 1) = 8/13 * 5/12 = (2/13 * 5/3) =10/39

P(X1 = 0, X2 = 0) = 8/13 * 7/12 =(2/13 * 7/3) = 14/39