Math 316 (Probability and Statistics) Joint Probability Distribution Problem

I’ll be posting some of my Math316 subject problems here and of course with its answer, since, I’m taking it up this semester! 😀

Problem # 1:

Two refills of ballpoint are to be selected at random from a box that contains 4 blue, 3 red and 2 black refills. If x is the number of blue and y is number of red refills selected, find:

• The joint probability distribution of x and y.
• P[(x,y)], where A is the region f(x+y) / x+y <=1.

Solution:

a.) The joint probability distribution of x&y.

f(0,0) = 4C0 . 3C0 . 2C2 / 9C2

= 1/36

f(0,1) = 4C0 . 3C1 . 2C1 / 9C2

= 1/6

f(1,0) = 4C1 . 3C0 . 2C1 / 9C2

= 2/9

f(1,1) = 4C1 . 3C1 . 2C0 / 9C2

= 1/3

f(2,0) = 4C2 . 3C0 . 2C0 / 9Cs

= 1/6

f(0,2) = 4C0 . 3C2 . 2C2 / 9C2

= 1/12

b.) P[(x,y)], where A is the region f(x+y) / x+y <=1.

P(A) = p(0,0) + P(0,1) + P(1,0)

= 1/36 + 1/6 + 2/9

= 5/12 (Since, x+y <=1)