Question

Give the probability of rolling a pair of fair dice such that:

a. both dice come up odd.

b. at least one die comes up odd.

c. exactly one die comes up odd.

d. the sum of the dice is odd.

Answer 1

Total possible outcomes when a pair of dice rolled = 36

Even numbers on a dice = 2,4,6

Odd numbers on a dice = 1,3,5

(a) Both pairs having odds = (1,1) (1,3) (1,5) (3,1) (3,3) (3,5) (5,1) (5,3) (5,5)

= 9 pairs

Required probability = 9/36 = 0.25​​​​​​

(b) pairs in which at least one dice comes up odd = all outcomes - all pairs with both dice even

Pairs with both even numbers = (2,2) (2,4) (2,6) (4,2) (4,4) (4,6) (6,2) (6,4) (6,6) = 9 pairs

Required pairs = 36 pairs - 9 pairs = 27 pairs

Required probability = 27/36 = 0.75​​​​​​

(c) exactly one die comes up odd = total pairs - all even pairs + all odd pairs

Required pairs = 36 - (9+9) = 18 pairs

Required probability = 18/36 = 0.50

(d) the sum of a pair of dice is only odd when we add an odd and an even number

So, the the pairs with 1 odd and 1 even numbers are just same as in the case (c) that is 18 pairs

Required probability = 18/36 = 0.50