Question

9. Two fair dice are rolled. Find the odds of rolling the following sums.(a) 2(b) 6(c) 9(d) Between 5 and 8 (exclusive; that is, not including 5 or 8)(e) Not more than 5

Answer 1

a. 1/36

b. 5/36

c. 1/9

d. 11/36

e. 5/18

EXPLANATION :

We can represent the outcome of rolling two fair dice as (x, y)

x = outcome of the first die

y = outcome of the second die

If we list the total number of outcomes, it will be :

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

It has a total of 6 x 6 = 36 possible outcomes.

So the probability of each question is the quotient of the required sum and the total possible outcomes which is 36.

a. Rolling a sum of 2 :

We can only have 1 possible sum and that is (1, 1)

So the probability is 1/36

b. Rolling a sum of 6 :

We can have :

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

A total of 5 possible outcomes.

So the probability is 5/36

c. Rolling a sum of 9 :

We can have :

(3, 6), (4, 5), (5, 4) and (6, 3)

A total of 4 possible outcomes.

So the probability is 4/36 or 1/9

d. Rolling a sum between 5 and 8, or having a sum of 6 and 7

Since we already have 5 possible outcome having a sum of 6 (letter b.)

Then for having a sum of 7 :

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

A total of 6 possible outcomes + 5 possible outcome for a sum of 6.

That's 11 possible outcomes.

So the probability is 11/36

e. Rolling a sum not more than 5, or having a sum of 2, 3, 4 and 5

For a sum of 2 :

(1, 1), only 1 possible outcome

For a sum of 3 :

(1, 2) and (2, 1)

2 possible outcomes

For a sum of 4 :

(1, 3), (2, 2) and (3, 1)

3 possible outcomes

For a sum of 5 :

(1, 4), (2, 3), (3, 2) and (4, 1)

4 possible outcomes

So the total possible outcome of having a sum not more than 5 is :

1 + 2 + 3 + 4 = 10

The probability is 10/36 or 5/18