A fair die is rolled. Consider events E = {1,3,5}, F = {2,3} and G = {2,3,4,5}

Find

(i) P(E|F) and P(F|E)

(ii) P(E|G) and P(G|E)

(iii) P((E ∪ F)|G) and P ((E ∩ F)|G)

The sample space for the given experiment will be:

S = {1, 2, 3, 4, 5, 6}

Here, E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5} ……….(I)

……….(II)

Now, E ∩ F = {3}, F ∩ G = {2, 3}, E ∩ G = {3, 5} ……….(III)

……….(IV)

(i) We know that

By definition of conditional probability,

[Using (II) and (IV)]

Similarly, we have

[Using (II) and (IV)]

(ii) We know that

By definition of conditional probability,

Similarly, we have

(iii) Clearly, from (I), we have

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

⇒ E ∪ F = {1, 2, 3, 5}

⇒ (E ∪ F) ∩ G = {2, 3, 5}

……….(V)

Now, we know that

By definition of conditional probability,

[Using (II) and (V)]

Similarly, we have E ∩ F = {3} [Using (III)]

And G = {2, 3, 4, 5} [Using (I)]

⇒ (E ∩ F) ∩ G = {3}

……….(VI)

So,

[Using (II) and (VI)]