Question

Two dice are thrown. Defined are the following two events A and B:

A={(x, y) : x + y = 9}, B = {(x, y) : x ≠ 3}, where (x, y) denote a point inthe sample space.

Check if events A and B are independent or mutually exclusive.

Answer 1

n(S) = number of element in sample space = 36

n(S) = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 1), (2, 2), (2, 3), (2, 4), (2,5), (2, 6) (3, 1), (3, 2),(3, 3,), (3, 4), (3, 5), (3, 6) (4, 1), (4, 2),(4, 3), (4, 4), (4, 5), (4, 6) (5, 1), (5, 2),(5, 3), (5, 4), (5, 5), (5, 6) (6, 1), (6, 2),(6, 3), (6, 4), (6, 5), (6, 6)}

A = Sum x + y = 9 

A = {(3, 6), (4, 5), (5, 4), (6, 3)}

n(A) = 4

B = {(x, y) : x ≠ 3} 

B = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6) (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6) (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6) (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

n(B) = 30

A ∩ B = {(4, 5), (5, 4), (6, 3)}

n(A ∩ B) = 3

P(A ∩ B) = `(n(A ∩ B))/(n(S))`

= `3/36`

= `1/12`

P(A) = `4/36`

 P(B) = `30/36`

`P(A) xx P(B) = 4/36 xx 30/36`

= `120/1296`

= `10/108`

= `5/54`

∴ P(AB) ≠ P(A) × P(B) (For independence and P(AB) ≠ 0)

Therefore, events A and B are neither independent nor mutually exclusive.