Question

 Cast a fair die and let X = 0 if 1, 2, or 3 spots appear, let X = 1 if 4 or 5 spots appear, and let X = 2 if 6 spots appear. Do this two independent times, obtaining X₁ and X₂. Calculate P(|X₁ – X₂| = 0).

Answer 1

|X1-X2|=0 means that both X1 and X2 is equal to:

(a) 0

(b) 1

(c) 2

(a) P((X1=0)⋂(X2=0)) = P(X1=0)*P(X2=0) = 0.5*0.5=0.25

(b) P((X1=1)⋂(X2=1))=P(X1=1)*P(X2=1)) = $({\frac 1 3 })^{2}={\frac 1 9}$

(c) P((X1=2)⋂(X2=2)) = P((X1=2)*P(X1=2) = $({\frac 1 6})^{2} = {\frac 1 {36}}$

P(|X1-X2|=0) $= 0.25 + {\frac 1 9 } + {\frac 1 {36}}={\frac 7 {18}}$