Question

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) = ?

Answer 1

`A.\ (1)/(3)\\B.\ (5)/(12)\\C.\ (7)/(36)\\`

Explanation:

Total outcomes possible: 36

A. Divisible by 3

Possible options are:

3, 6, 9 and 12.

Possible outcomes for 3 are: {(1,2), (2,1)} Count 2

Possible outcomes for 6 are: {(1,5), (2,4), (3,3), (5,1),(4,2)} Count 5

Possible outcomes for 9 are: {(3,6), (4,5), (5,4),(6,3)} Count 4

Possible outcomes for 12 are: {(6,6)} Count 1

Total count = 2 + 5 + 4 + 1 = 12

Probability of an event E can be formulated as:

`P(E) = \frac{\text{Number of favorable cases}}{\text {Total number of cases}}`

`P(A) = (12)/(36) = (1)/(3)`

B. Less than 7:

Possible sum can be 2, 3, 4, 5, 6

Possible cases for sum 2: {(1,1)} Count 1

Possible cases for sum 3: {(1,2), (2,1)} Count 2

Possible cases for sum 4: {(1,3), (3,1), (2,2)} Count 3

Possible cases for sum 5: {(1,4), (2,3), (3,2),(4,1)} Count 4

Possible cases for sum 6: {(1,5), (2,4), (3,3), (5,1),(4,2)} Count 5

Total count = 1 + 2 + 3 + 4 + 5 = 15

`P(B) = (15)/(36) = (5)/(12)`

C. Divisible by 3 and less than 7:

`P(A \cap B) = \frac{n(A\cap B)}{\text{Total Possible outcomes}}`

Here, common cases are:

Possible outcomes for 3 are: {(1,2), (2,1)} Count 2

Possible outcomes for 6 are: {(1,5), (2,4), (3,3), (5,1),(4,2)} Count 5

`P(A \cap B) = \frac{7}{\text{36}}`