10 points] Lizzie rolls two dice. What is the probability that the sum of the dice is:
(a) divisible by 3?

(b) less than 7?

(c) divisible by 3 and less than 7?

1. Probability = ⅓

2. Probability = 5/12

3. Probability = 7/36

Step-by-step explanation:

Given

Two dice

Required

Probability of getting a sum

- divisible by 3

- less than 7

- divisible by 3 and less than 7

To solve this; first, we need to list out the sample space.

Let the first die be represented by S1 and the second be represented by S2

S1 = {1,2,3,4,5,6}

S2 = {1,2,3,4,5,6}

The sample space which is the sum of both dice is as follows

S = {(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)}

S = {2,3,4,5,6,7,3,4,5,6,7,8,4,5,6,7,8,9,5,6,7,8,9,10,6,7,8,9,10,11,7,8,9,10,11,12}

Total outcome = 36

To calculate the probability of getting a sum divisible by 3. We first list out all sums divisible by 3. Let that be represented by T

T = {3,6,3,6,6,9,6,9,6,9,9,12}

The total outcome of this is 12.

Hence the probability of obtaining a sum divisible by 3 = 12/36

Probability = ⅓

Tot calculate the probability of obtaining a sum less than 7. We first list out all sums less than 7. Let that be represented by T

T = {2,3,4,5,6,3,4,5,6,4,5,6,5,6,6}

The total outcome of this is 15

Hence the probability of obtaining a sum less than 7 = 15/36

Probability = 5/12

To get the probability of a sum divisible by 3 and less than 7. We first list out all sums divisible by 3 and at the same time less than 7. Let that be represented by T

T = {3,6,3,6,6,6,6}

The total outcome of is 7

Hence the probability of obtaining a sum divisible by 3 and less than 7 = 7/36

Probability = 7/36