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)P1P2
(ii)(1-P1-2) P2
(iii)1-(1-P1) (1-P2)
(iv)P1+P2-2P1P2
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:
(A ∪ B)’ = A’∩ B’ |
=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.