Question

1. Consider the following joint probability table: B 0 CIPLA,B,C) 0 A 0.03 0 0.15 0.02 0 0.11 0.14 1 0.28 0.09 1 0.18 0 0 0 0 0 a) What is P(A-1, B-0, C-1)? b) What is P(B-1)? c) What is P(A 0, C 1)? d) What is P(A-1 or B-0)? e) What is P(A 1|B 0)? f) If C 1 (consider only the last four columns of the table), is A independent of B? In other words, does P(A,BIC-1) PAIC-1) P(BIC-1)? g) If B-0, is A independent of C?

Only f）& g） Thanks

Answer 1

Here we have to solve only part (f) and (g)

(f) Here if C = 1

now we have to find that is A independent of B or in other words

P(A , B l C = 1) = P(A l C = 1) P(B l C = 1)

so here we will take values like where A = 1 and B = 1

P(A = 1, B = 1 l C = 1) = 0.09

and

P(A =1 l C = 1) = 0.14 + 0.09 = 0.23

P(B =1 l C = 1) = 0.09 + 0.18 = 0.27

now multiplying it

P(A =1l C = 1) * P(B =1 l C = 1) = 0.23 * 0.27 = 0.0621 $\neq$ 0.09

so here A and B are not independent when C = 1.

(g) Now B = 0 so now we have to identify that A and C are indepent

now we have to find that is A independent of C or in other words

P(A , C l B = 0) = P(A l B = 0) P(C l B = 0)

so here we take one value where A = 1 and C = 1

P(A = 1, C = 1 l B = 1) = 0.14

and

P(A =1 l B = 0) = 0.14 + 0.03 = 0.17

P(C =1 l B = 0) = 0.14 + 0.28 = 0.42

now multiplying it

P(A =1 l B = 0) * P(C =1 l B = 0) = 0.17 * 0.42 = 0.00714 $\neq$ 0.14

so here A and C are not independent when B = 0.