Question

Let E1 and E2 be two independent events such that p(E1) = p1 and P(E2) = p2. Describe in words of the events whose probabilities are: (i) p1 p2 (ii) (1–p1) p2 (iii) 1 – (1 – p1)(1 – p2) (iv) p1 + p2 – 2p1p2 Select the correct answer from above options

Answer 1

Given that P(E1) =P1 and P(E2) =P2 (i)P1P2 =P(E1). P(E2) =P (E1 Ո E2) So, E1 and E2 occur simultaneously. (ii)(1-P1) P2 As we know P(A)+P(A)’ = 1 =P(E1)’. P(E2) = P (E1’ Ո E2) So, E1 does not occur but E2 occur. (iii)1-(1-P1) (1-P2) As we know P(A)+P(A)’ = 1 =1-P(E1)’P(E2)’ =1-P (E1’ Ո E2’) By De Morgan’s laws: =1-P (E1 Ս E2)’ As we know P(A)+P(A)’ = 1 = 1- [1-P (E1 Ս E2)] = P (E1 Ս E2) So, either E1 or E2 or both E1 and E2 occurs. (iv)P1+P2-2P1P2 = P(E1) + P(E2) – 2P(E1) P(E2) = P(E1) + P(E2) – 2P (E1 Ո E2) = P (E1 Ս E2)- 2P (E1 Ո E2) So, either E1 or E2 occurs but not both.