Question

Two dice are thrown.

Let A be the event that an odd number is obtained on the first dice, B be the event that the number obtained is greater than 5 on the first dice, C be the event that the number obtained on the second dice is smaller than 5, and D be the event that the sum of the two numbers obtained is 8.

State whether each of the following is a pair of independent events or dependent events.

(a) A and B

(b) A and C

(c) B and D

(d) C and D

Answer 1

(a)

A and B

P(A) = 3/6 = 1/2 (odd numbers are 1, 3, 5)

P(B) = 1/6 (Number greater than 5 is only 6)

P(A and B) = P(odd number and greater than 5 on the first dice) = 0 (No odd number is greater than 5)

Since P(A and B) P(A) P(B), A and B are dependent events.

(b)

A and C

P(A) = 3/6 = 1/2 (odd numbers are 1, 3, 5)

P(C) = 4/6 = 2/3 (Number smaller than 5 are 1, 2, 3, 4)

P(A and C) = P(odd number of 1st dice and Number smaller than 5 on 2nd dice) = P(A) * P(C) (As they are outcomes of different dices)

Thus, A and C are independent events.

(c)

B and D

P(B) = 1/6 (Number greater than 5 is only 6)

P(D) = 5/36 (Combinations of numbers where sum is 8 out of 36 combinations is, (6,2), (5,3), (4,4) , (3,5), (2.6))

P(B and D) = P(Number greater than 5 on first dice and sum is 8 ) = P(first dice is 6 and second dice is 2) = 1/36   

Since P(B and D) P(B) P(D), B and D are dependent events.

(d)

C and D

P(C) = 4/6 = 2/3 (Number smaller than 5 are 1, 2, 3, 4)

P(D) = 5/36 (Combinations of numbers where sum is 8 out of 36 combinations is, (6,2), (5,3), (4,4) , (3,5), (2.6))

P(C and D) = P(Number smaller than 5 on second dice and sum is 8 ) = P(first dice is 6 and second dice is 2) + P(first dice is 5 and second dice is 3) + P(first dice is 4 and second dice is 4) = 3/36   

Since P(C and D) P(C) P(D), C and D are dependent events.