Question

P(A) = 0.3, P(B) = 0.7 (a) Can you compute P(A and B) if you only know P(A) and P(B)?(b) 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|B)?(c) 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)?

Answer 1

(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,

`P(A|B)=(P(A and B))/(P(B)) = (0.1)/(0.7) = 0.143`