If a fair die is rolled, then the sample space is S = {1, 2, 3, 4, 5, 6}
Now, E ∩ F = F ∩ E = {3}, E ∩ G = G ∩ E = {3, 5}, E ∪ F = {1, 2, 3, 5},
(E ∪ F) ∩ G = {2, 3, 5}, (E ∩ F) ∩ G = {3}.
P(E) = 3/6, P(F) = 2/6, P(G) = 4/6, P(E ∩ F) = 1/6, P(E ∩ G) = 2/6, P(E ∪ F) = 4/6, P((E ∪ F) ∩ G) = 3/6, P((E ∩ F) ∩ G) = 1/6
(i) P (E | F) =

P(E∩F)

P(F)

=

1∕6

2∕6

=

1

2

and P(F | E) =

P(E∩F)

P(E)

=

1∕6

3∕6

=

1

3

(ii) P(E | G) =

P(E∩G)

P(G)

=

2∕6

4∕6

=

1

2

and P(G | E) =

P(G∩E)

P(E)

=

2∕6

3∕6

=

2

3

(iii) P ((E ∪ F) | G) =

P((E∪F)∩G)

P(G)

=

3∕6

4∕6

=

3

4

and P((E ∩ F) | G) =

P((E∩F)∩G)

P(G)

=

1∕6

4∕6

=

1

4