Question

Two refills for a ball point pen are selected at random for a box that contains 55 blue, 3 red and 3 green refills. If X is the number of blue refills and Y is the number of red refills selected .Find the joint probability function f (x, y) and p[(x,y)∈A ], where A is the region {(x,y): x+y<=1}

Answer 1

Part 1

The possible pairs of values (x, y) are (0, 0), (0, 1), (1, 0), (1, 1), (0, 2), and (2, 0), where p(0, 1), for

example represents the probability that a red and a green refill are selected. The total number of equally likely ways of selecting any 2 refills from the 8 is

$\begin{pmatrix} 61 \\ 2 \end{pmatrix}= \frac{61!}{2!58!}=107970$

The number of ways of selecting 1 red from 3 red refills and 1 green from 3 green refills is

$\begin{pmatrix} 3 \\ 1 \end{pmatrix}\begin{pmatrix} 3 \\ 1 \end{pmatrix}=9$

Hence, $p(1, 1) = \frac{9}{107970}$ . Similar calculations yield the probabilities for the other cases, which are presented in the following table. Note that the probabilities sum to 1.

$p(x,y)= \frac{\begin{pmatrix} 55 \\ x \end{pmatrix}\begin{pmatrix} 3 \\ y \end{pmatrix}\begin{pmatrix} 3 \\ 3-x-y \end{pmatrix}}{\begin{pmatrix} 61 \\ 2 \end{pmatrix}}\\ For \space x =0,1,2\\ For \space y =0,1,2\\ 0≤x+y≤2$

Part 2

$P[(X, Y) \in A] = P(X + Y≤1)\\ = p(0, 0) + p(0, 1) + p(1, 0)\\ =\frac{55}{107970}+\frac{3}{53985}+\frac{165}{107970}\\ =\frac{113}{53985}$