Question

Two dice are thrown simultaneously. What is the probability that
(i) 5 will not come up on either of them?
(ii) 5 will come up on at least one of them?
(iii) 5 will come up at both the dice?

Answer 1

When two dice are thrown simultaneously, the possible outcomes are:

(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)
Number of all possible outcomes = 36

(i) Let E₁ be the event in which 5 will not come up on either of them.
Then, the favourable outcomes are:
(1,1), (1,2), (1,3), (1,4), (1,6), (2,1), (2,2), (2,3), (2,4), (2,6), (3,1), (3,2), (3,3), (3,4), (3,6), (4,1), (4,2), (4,3), (4,4), (4,6), (6,1), (6,2), (6,3), (6,4) and (6,6).
Number of favourable outcomes = 25
∴ P (5 will not come up on either of them) = P (E₁) = $\frac{25}{36}$

(ii) Let E₂ be the event in which 5 will come up on at least one.
Then the favourable outcomes are:
(1,5), (2,5), (3,5), (4,5), (5,5), ​(6,5), ​(5,1), ​(5,2), ​​(5,3), ​​(5,4) and ​​(5,6)
Number of favourable outcomes = 11
∴ P ( 5 will come up on at least one) = P (E₂) = $\frac{11}{36}$

(iii) Let E₃ be the event in which 5 will come up on both the dice.
Then the favourable outcome is (5,5).
Number of favourable outcomes = 1
∴ P (5 will come up on both the dice​) = P (E₂) = $\frac{1}{36}$