Chapter 3.1, Problem 20GP

a.

To determine

Find the probability that when rolling two dice the sum of the dice is not 6.

Answer to Problem 20GP

The probability that when rolling two dice the sum of the dice is not 6 is $\frac{31}{36}$.

Explanation of Solution

Calculation:

The possible outcomes when rolling two dice is given below:

$\left\{\begin{array}{l}\left(1,1\right),\left(1,2\right),\left(1,3\right),\left(1,4\right),\left(1,5\right),\left(1,6\right),\\ \left(2,1\right),\left(2,2\right),\left(2,3\right),\left(2,4\right),\left(2,5\right),\left(2,6\right),\\ \left(3,1\right),\left(3,2\right),\left(3,3\right),\left(4,4\right),\left(3,5\right),\left(3,6\right),\\ \left(4,1\right),\left(4,2\right),\left(4,3\right),\left(4,4\right),\left(4,5\right),\left(4,6\right),\\ \left(5,1\right),\left(5,2\right),\left(5,3\right),\left(5,4\right),\left(5,5\right),\left(5,6\right),\\ \left(6,1\right),\left(6,2\right),\left(6,3\right),\left(6,4\right),\left(6,5\right),\left(6,6\right)\end{array}\right\}$

The sum of outcomes is given below:

$S=\left\{\begin{array}{l}2,3,4,5,6,7,\\ 3,4,5,6,7,8,\\ 4,5,6,7,8,9,\\ 5,6,7,8,9,10,\\ 6,7,8,9,10,11,\\ 7,8,9,10,11,12\end{array}\right\}$

The possible number of outcomes for getting 6 is 5 and the total number of outcomes is 36.

Therefore, the number of outcomes for not getting 6 is 31 $\left(=36-5\right)$.

The required probability is obtained as follows:

$\begin{array}{c}\text{Required probability}=\frac{\text{Favorable number of outcomes for not getting a sum 6}}{\text{Total number of outcomes}}\\ =\frac{31}{36}\end{array}$

Thus, the probability that when rolling two dice the sum of the dice is not 6 is $\frac{31}{36}$.

b.

To determine

Find the probability that when rolling two dice the sum is at least 4.

Answer to Problem 20GP

The probability that when rolling two dice the sum is at least 4 is $\frac{11}{12}$.

Explanation of Solution

Calculation:

From Part (a), it can be observed that the possible number of outcomes for getting the sum that is at least 4 is 33.

The total number of outcomes is 36.

The required probability is obtained as follows:

$\begin{array}{c}\text{Required probability}=\frac{\text{Favorable number of outcomes for getting sum is at least 4}}{\text{Total number of outcomes}}\\ =\frac{33}{36}\\ =\frac{11}{12}\end{array}$

Thus, the probability of $P\left(A^c\right)\text{ is }\frac{1}{36}.$

c.

To determine

Find the probability that when rolling two dice the sum is no more than 10.

Answer to Problem 20GP

The probability that when rolling two dice the sum is no more than 10 is $\frac{11}{12}$.

Explanation of Solution

Calculation:

From Part (a), it can be observed that the possible number of outcomes for getting the sum that is no more than 10 is 33. That is, the number of outcomes for getting the sum that is less than or equal to 10 is 33.

The required probability is obtained as follows:

$\begin{array}{c}\text{Required probability}=\frac{\text{Favorable number of outcomes for getting sum is no more than 10}}{\text{Total number of outcomes}}\\ =\frac{33}{36}\\ =\frac{11}{12}\end{array}$

Thus, the probability that when rolling two dice the sum is no more than 10 is $\frac{11}{12}$.