Answer:
a) f(x,y) =[tex]\frac{\binom{3}{x}\binom{2}{y}\binom{3}{2-x-y}}{\binom{8}{2}}[/tex] ; x = 0, 1 , 2; y = 0, 1 , 2; 0 ≤ x+y ≥ 2
b) = [tex]\frac{9}{14}[/tex]
Step-by-step explanation:
joint probability is a function that characterizes the distribution of a random variable. If X and Y be two random variables then the joint probability will be P(X = x, Y=y)
Given Data,
X = The number of blue Pens
Y = The number of red Pens
a)
possible outcomes(X, Y) are (0, 0), (0, 1), (1, 0), (1, 1), (0, 2), (2,0)
Please refer fig. also
total number ways of selecting any 2 pens = [tex]\binom{8}{2}[/tex]= [tex]\frac{8!}{2! 6!}[/tex] =28
f(x,y) = [tex]\frac{\binom{3}{x}\binom{2}{y}\binom{3}{2-x-y}}{\binom{8}{2}}[/tex] ; x = 0, 1 , 2; y = 0, 1 , 2; 0 ≤ x+y ≥ 2
b)
P(X,Y)∈A = P(X + Y ≤ 1)
= P(0,0) + P(1,0) + P(0,1)
= [tex]\frac{3}{28}[/tex] + [tex]\frac{3}{14}[/tex] + [tex]\frac{9}{28}[/tex]
= [tex]\frac{9}{14}[/tex]
