P(a) = 0.3, p(b) = 0.7 (a) can you compute p(a and b) if you only know p(a) and p() assuming that events a and b arise from independent random processes, i. what is p(a and b)? ii. what is p(a or b)? iii. what is p(a|) if we are given that p(a and b) = 0.1, are the random variables giving rise to events a and b indepen-dent? (d) if we are given that p(a and b) = 0.1, what is p(a|b)?

(a) No, we can't compute P(A and B) because we don't know the A and B are independent or not. Also If it not independent we don't know the value of P(A or B)

(b) (i) If A and B are independent then,

P(A and B) = P(A) × P(B)

⇒ P(A and B) = 0.3 × 0.7 = 0.21

(ii) By, Additional theorem of Probability,

P(A or B) = P(A) + P(B) - P(A and B)

⇒ P(A or B) = 0.3 + 0.7 - 0.21 = 0.79

(iii) P(A|B) = P(A) = 0.3

(c) For the events to be independent, it should follow:

P( A and B) = P(A) × P(B)

0.1 = 0.3 × 0.7

⇒ 0.1 ≠ 0.21

Thus, A and B are not independent event.

(d) By the Conditional Probability definition, we know that,

answer 1 : option "c"

answer 2 : option "d"

answer 3 : option "b"

answer 4 : option "d"

answer 5 : option "b"

answer 6 : option "a"

answer 7 : option "c"

answer 8 : option "d"

answer 9 : option "c"

answer 10 : option "d"

answer 11 : option "c"

answer 12 option "b"

answer 13 option "c"

answer 14 option "c"

answer 15 option "d"

answer 16 option "a"

answer 17 option "b"

answer 18 option "c"

answer 19 option "a"

answer 20 option "a"

answer 21 option "c"

d