Question

Let `E_(1)` and `E_(2)` be two events such that `P(E_(1))=0.3, P(E_(1) uu E_(2))=0.4` and `P(E_(2))=x`. Find the value of x such that (i) `E_(1)` and `E_(2)` are mutually exclusive, (ii) `E_(1)` and `E_(2)` are independent. Select the correct answer from above options

Answer 1

(i) Let `E_(1)` and `E_(2)` be mutially exclusive. Then `E_(1) nn E_(2)= varphi`. `:. P(E_(1) uu E_(2))=P(E_(1))+P(E_(2))` `implies 0.4=0.3+x` `implies x=0.1` Thus, when `E_(1)` and `E_(2)` mutually exclusive, then `x=0.1`. (ii) Let `E_(1)` and `E_(2)` be two independent events. Then. `P(E_(1) nn E_(2))=P(E_(1))xxP(E_(2))=0.3xx x=0.3 x`. `:. P(E_(1) uu E_(2))=P(E_(1))+P_(E_(2))-P(E_(1) nn E_(2))` `implies 0.4=0.3+x-0.3 x` `implies 0.7 x=0.1` `implies x=0.1/0.7=1/7` Thus, when `E_(1)` and `E_(2)` are independent, then `x=1/7`.