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?
(iii) 5 will come up at both dice?

Answer 1

I will give the pic of answer
hope it help you

Answer 2

The easiest and most basic way to solve this problem is to start by writing the sample space.

For one dice, the sample space would have 6 items. For two dice, the sample space would have 6² = 36 items

The sample space is as follows:

S = { (6,6), (6,5), (6,4), (6,3), (6,2), (6,1),

       (5,6), (5,5), (5,4), (5,3), (5,2), (5,1),

       (4,6), (4,5), (4,4), (4,3), (4,2), (4,1),

       (3,6), (3,5), (3,4), (3,3), (3,2), (4,1),

       (2,6), (2,5), (2,4), (2,3), (2,2), (2,1),

       (1,6), (1,5), (1,4), (1,3), (1,2), (1,1)}

(i) Probability that 5 will not come up on either of them:

Checking the sample space above, we will find that:

S(A) = { (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), (6,6)}

Therefore, P(A) = $\frac{25}{36}$

(ii) Probability that 5 will come up on at least one:

This means that 5 can come up on either one or both of the dice.

Checking the original sample space, we will find that:

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

Therefore, P(B) = $\frac{11}{36}$

(iii) Probability that 5 will come up at both dice:

Checking the original sample space, we will find that:

S(C) = {(5,5)}

Therefore, P(C) = $\frac{1}{36}$

Hope this helps :)