How can one compute the joint probability of A and B with the information provided?

Question

Let $(A, B)$ be a discrete R.V. (vector) with the set {$(0,0),(0,1),(1,0),(1,1)$} Their marginal distributions are $A∼Binomial(1, \frac{1}{2})$ and $B∼Bin(1, \frac{1}{2})$. Given that $P(A=B)=0.8$. Find $P(AB = 0)$.

Answer 1

You have $ P(A=0,B=0)+P(A=0,B=1) +P(A=1,B=0)+P(A=1,B=1)=1 $ This gives you $ P(A=0,B=1) +P(A=1,B=0) +P(A=B)=1, $ thus using $P(A=B)=0.8$, $$P(A=0,B=1) +P(A=1,B=0)=0.2.$$ Also $0.5=P(A=1)= P(A=1,B=1)+P(A=1,B=0)$ and $0.5=P(B=1)= P(A=0,B=1)+P(A=1,B=1)$, summing the last two equations get $1=2P(A=1,B=1)+P(A=1,B=0)+P(A=0,B=1)$, plug in the value $P(A=0,B=1) +P(A=1,B=0)=0.2$ and get $P(A=1,B=1)=0.4$. Now $$P(AB=0)=1-P(A=1,B=1)= 0.6.$$