Question

A fair die is rolled. Consider events E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5} Find P ((E ∪ F)|G) and P ((E ∩ G)|G)

Answer 1

E = {1, 3, 5}, F = {2, 3}, G = {2, 3, 4, 5}

E ∩ G = {3, 5}, F ∩ G = {2, 3}, (E ∩ F) ∩ G = {3}

P(E ∩ G) = `2/6`, P(F ∩ G) = `2/6`, P[(E ∩ F) ∩ G] = `1/6`

Now P(E ∪ F|G) = P(E|G) + P(F|G) − P[(E ∩ F) ∩ G]

=`(P(E ∩ G))/(P(G)) + (P(E ∩ G))/(P(G)) - (P[(E ∩ F) ∩ G])/(P(G))`  

= `(2/6 ÷ 4/6) + (2/6 ÷ 4/6) - (1/6 ÷ 4/6)`

= `2/4 + 2/4 - 1/4`

= `3/4`

P(E ∩ F|G) = `(P[(E ∩ F) ∩ G])/(P(G))`

= `1/6 ÷ 4/6`

= `1/4`