Question

Two events ?1 and ?2 are defined on the same probability space such that: ?(?1 ) = 0.7 , ?(?2 ) = 0.5 , and ?(?1 ??? ?2 ) = 0.3. a) Find ?(?1 ?? ?2 ). b) Find ?(?1 | ?2 ). c) Are ?1 and ?2 mutually exclusive (disjoint)? and why? d) Are ?1 and ?2 independent? and why?

Answer 1

Solution:

We are given two events ?1 and ?2 defined on the same probability space such that:

P(E₁) =0.7, P(E₂) =0.5 and P(E₁ and E₂) =0.3

Part a) We have to find P(E₁ or E₂) =............?

Thus using addition rule of probability:

P(E₁ or E₂) =P(E₁) + P(E₂) - P(E₁ and E₂)

P(E₁ or E₂) = 0.7 + 0.5 - 0.3

P(E₁or E₂) = 0.9

Part b) Find conditional probability:  P(E₁ | E₂) =..........?

Use following conditional rule of probability:

Part c) Are ?1 and ?2 mutually exclusive (disjoint)?

Events ?1 and ?2 are mutually exclusive (disjoint) if:

P(E1 and E2) = 0

but P(E₁ and E₂) =0.3 which is not equal to 0, thus events E₁ and E₂ are not mutually exclusive.

Part d) Are ?1 and ?2 independent? and why?

Events are independent if and only if, P(E₁ | E₂) = P(E₁)

Since and

that is: , thus events E₁ and E₂ are not independent events.