Chapter 6.8, Problem 18E

i.

To determine

To find the different possible outcomes.

Answer to Problem 18E

The number of different outcomes that are possible for the pairs of number rolled is $32$ .

Explanation of Solution

Given:

There is a $8$ sided number octahedron and a $4$ sided number pyramid.

Formula Used:

If one event can occur in $m$ ways, and for each of these ways a second event can occur in $n$ ways, then the number of ways that the two events can occur together is $m\times n$ .

Calculation:

The number of different outcomes that are possible is:

  $=$ Number of outcomes for the octahedron $\times$ Number of outcomes for the pyramid

  $\begin{array}{l}=8\times 4\\ =32\end{array}$

Hence,

The number of different outcomes that are possible for the pairs of number rolled is $32$ .

ii.

To determine

To find the probability.

Answer to Problem 18E

The probability is $\frac{1}{8}$ to get the same number.

Explanation of Solution

Given:

There is a $8$ sided number octahedron and a $4$ sided number pyramid.

Formula used:

  $P\left(\text{event}\right)=\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}$

Calculation:

Total number of possibleoutcomes $=32$ .

Total number of favorable outcomes of rolling the same number is $\left(1,1\right)\left(2,2\right)\left(3,3\right)\left(4,4\right)=4$

Therefore,

  $\begin{array}{l}P\left(\text{event}\right)=\frac{\text{Number of favorable outcomes}}{\text{Number of possible outcomes}}\\ P=\frac{4}{32}=\frac{1}{8}\end{array}$

Hence,

The probability is $\frac{1}{8}$ to get the same number.

iii.

To determine

To find the odds in favor.

Answer to Problem 18E

The odds in favor is $\frac{1}{1}$ or $1\text{ to 1}$ of rolling a number greater than the $4$ sided number pyramid.

Explanation of Solution

Given:

There is a $8$ sided number octahedron and a $4$ sided number pyramid.

Formula used:

  $\text{Odds in favor}=\frac{\text{Number of favorable outcomes}}{\text{Number of unfavorable outcomes}}$

Calculation:

Total number of outcomes $=32$

Number of favorable outcomes of rolling a number greater than the $4$ sided number

Pyramid $=16$

Number of unfavorable outcomes of rolling a number greater than the $4$ sided number pyramid

  $=$ Total outcomes $-$ Number of favorable outcomes

  $\begin{array}{l}=32-16\\ =16\end{array}$

Therefore,

  $\text{Odds in favor}=\frac{\text{Number of favorable outcomes}}{\text{Number of unfavorable outcomes}}=\frac{16}{16}=\frac{1}{1}$

Hence,

The odds in favor is $\frac{1}{1}$ or $1\text{ to 1}$ of rolling a number greater than the $4$ sided number pyramid.