Question

An unbiased die is thrown twice. Let the event A be the odd number on the first throw and B the event odd number on the second throw. Check whether A and B events are independent.

Answer 1

When a die is thrown twice, the sample space is 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)}

n(S) = 36

The event A is odd number on the first throw

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

n(A) = 18

P(A) = `18/36 = 1/2`

The event B is odd number on the second throw.

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

n(B) = 18

P(B) = `18/36 = 1/2`

A ∩ B = {(1, 1), (1, 3), (1, 5), (3, 1), (3, 3), (3, 5), (5, 1), (5, 3), (5, 5)}

n(A ∩ B) = 9

P(A ∩ B) = `9/36 = 1/4`

Also P(A) × P(B) = `1/2 xx 1/2 = 1/4`

Thus P(A ∩ B) = P(A) × P(B)

∴ A and B are independent events.