Question

An unbiased die is thrown twice. Let the event A be an 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) = \(\frac{18}{36}=\frac{1}{2}\)

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) = \(\frac{18}{36}=\frac{1}{2}\)

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) = \(\frac{9}{36}=\frac{1}{4}\)

Also P(A).P(B) = \(\frac{1}{2}.\frac{1}{2}=\frac{1}{4}\)

Thus P(A ∩ B) = P(A).P(B) 

∴ A and B are independent events.