Question

Two refills for a ball point pen are selected at random for a box that contains ‘a’

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) element A],

where A is the region { (x,y): x+y <=1}.

Answer 1

Total # of ways to choose 2 refills $=9C_2=\frac{9!}{\left(2!\times \left(9-2\right)!\right)}=36$

$P(X = 0, Y = 0) = \left(no\:blue\:and\:no\:red\right) = P\left(all\:are\:green\right)$ $=\frac{3C_2}{36}=\frac{3}{36}=\frac{1}{12}$

$P\left(X\:=\:0,\:Y\:=\:1\right)\:=\:P\left(1\:red,\:1\:green\right)\:=\:\frac{3C_1\:\times 3C_1}{36}=\:\frac{3\times 3}{36}=\frac{1}{4}$

$P\left(X\:=\:0,\:Y\:=\:2\right)\:=\:P\left(2\:red\right)\:=\:\frac{3C_2}{36}\:=\:\frac{3}{36}\:=\frac{1}{12}$

$\:P\left(X\:=\:1,\:Y\:=\:0\right)\:=\:P\left(1\:blue,\:1\:green\right)\:=\frac{3C_1\:\times 3C_1}{36}=\:\frac{3\times 3}{36}=\frac{1}{4}$

$P\left(X\:=\:1,\:Y\:=\:1\right)\:=\:P\left(1\:blue,\:1\:red\right)\:=\frac{3C_1\:\times 3C_1}{36}=\:\frac{3\times 3}{36}=\frac{1}{4}$

$P\left(X\:=\:1,\:Y\:=\:2\right)\:=\:P\left(1\:blue,\:2\:red\right)=\:0$

$P\left(X\:=\:2,\:Y\:=\:0\right)\:=\:P\left(2\:blue\right)\:=\frac{\:3C_2}{36}\:\:=\:\frac{3}{36}\:=\frac{1}{12}$

$P\left(X\:=\:2,\:Y\:=\:1\right)=P\left(2\:blue,\:1\:red\right)=0$

$P\left(X\:=\:2,\:Y\:=\:2\right)=P\left(2\:blue,\:2\:red\right)=0$