Chapter 13.6, Problem 15PPS

(a)

To determine

To calculate: To make a table showing the sample space

Answer to Problem 15PPS

Sample space is as follows:

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

Explanation of Solution

Given information: The below spinners are each spun once

  

Formula Used:

Sample space of an experiment is the set of all possible outcomes of that experiment

Calculation:

Given that spinners are each spun once

  

Here, experiment is that each spinners are spun once

Total possible outcomes of experiment $=4\times 4=16$

Sample space is as follows:

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

Conclusion:

Hence, sample space of experiment is listed

(b)

To determine

To calculate: To find the probability of even sum

Answer to Problem 15PPS

Theprobability of even sum is $\frac{1}{2}$

Explanation of Solution

Given information: The below spinners are each spun once

  

Formula Used:

Probability is defined as the likelihood of an event.

Probability is calculated as

  $\mathrm{Probability}=\frac{Event}{Outcomes}$

Calculation:

Given that spinners are each spun once

  

Here, experiment is that each spinners are spun once

Total possible outcomes of experiment $=4\times 4=16$

Sample space is as follows:

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

Now, from above sample, even sum is possible for below outcomes

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

Thus, number of outcomes for even sum $=8$

Thus, probability of even sum is calculated as

  $\begin{array}{l}P\left(evensum\right)=\frac{8}{16}\\ P\left(evensum\right)=\frac{1}{2}\end{array}$

Conclusion:

Hence, $P\left(evensum\right)=\frac{1}{2}$

(c)

To determine

To calculate: To find the probability of two even numbers

Answer to Problem 15PPS

Theprobability of two even numbers is $\frac{1}{4}$

Explanation of Solution

Given information: The below spinners are each spun once

  

Formula Used:

Probability is defined as the likelihood of an event.

Probability is calculated as

  $\mathrm{Probability}=\frac{Event}{Outcomes}$

Calculation:

Given that spinners are each spun once

  

Here, experiment is that each spinners are spun once

Total possible outcomes of experiment $=4\times 4=16$

Sample space is as follows:

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

Now, from above sample, two even numbers is possible for below outcomes

  $\begin{array}{l}\left(2,2\right),\left(2,4\right)\\ \left(4,2\right),\left(4,4\right)\end{array}$

Thus, number of outcomes for two even numbers $=4$

Thus, probability of two even numbers is calculated as

  $\begin{array}{l}P\left(twoevennumbers\right)=\frac{4}{16}\\ P\left(twoevennumbers\right)=\frac{1}{4}\end{array}$

Conclusion:

Hence, $P\left(twoevennumbers\right)=\frac{1}{4}$