Chapter 5, Problem 92DE

a.

To determine

1. Find the probability that the home is in township 1or has a pool.

2. Find the probability that the home has a pool given that it is in township 3.

3. Find the probability that the home has a pool and is in township 3.

Answer to Problem 92DE

1. 1. The probability that the home is in township 1or has a pool is 0.72.
2. 2. The probability that the home has a pool given that it is in township 3 is 0.72.
3. 3. The probability that the home has a pool and is in township 3 is 0.17.

Explanation of Solution

Calculation:

The below table shows the number of homes that have a pool, versus the number of homes that do not have a pool, in each of the five townships.

	Township
pool	1	2	3	4	5	Total
Yes	6	12	18	18	13	67
No	9	8	7	11	3	38
Total	15	20	25	29	16	105

Home is in township 1 or has a pool:

The probability that the home is in township 1 or has a pool is obtained as follows:

$\begin{array}{c}P\left(\begin{array}{l}\text{home is in township 1 }\\ \text{or has a pool}\end{array}\right)=\frac{\left(\begin{array}{l}\text{Number of homes that have pool +}\\ \text{Number of homes that are in township 1}-\\ \text{Number of homes that have pool and is in township 1 }\end{array}\right)}{\text{Total number of homes}}\\ =\frac{67+15-6}{105}\\ =\frac{76}{105}\\ =0.72\end{array}$

Thus, the probability that the home is in township 1 or has a pool is 0.72.

Home has a pool given that it is in township 3:

The probability that the home has a pool given that it is in township 3 is obtained as follows:

$\begin{array}{c}P\left(\text{home has a pool given that it is in township 3}\right)=\frac{\left(\begin{array}{l}\text{Number of homes that have pool }\\ \text{and is in township 3 }\end{array}\right)}{\text{Total number of homes in township 3}}\\ =\frac{18}{25}\\ =0.72\end{array}$

Thus, the probability that the home has a pool given that it is in township 3 is 0.72.

Home has a pool and is in township 3:

The probability that the home has a pool and is in township 3 is obtained as follows:

$\begin{array}{c}P\left(\text{home has a pool and is in township 3}\right)=\frac{\left(\begin{array}{l}\text{Number of homes that have pool }\\ \text{and is in township 3 }\end{array}\right)}{\text{Total number of homes}}\\ =\frac{18}{105}\\ =0.17\end{array}$

Thus, the probability that the home has a pool and is in township 3 is 0.17.

b.

To determine

1. Find the probability that the home has a garage attached.

2. Find the probability that the home does not have a garage attached given that it is township 5.

3. Find the probability that the home has a garage attached and is in township 3.

4. Find the probability that the home does not have a garage attached or is in township 2.

Answer to Problem 92DE

1. 1. The probability that the home has a garage attached is 0.68.
2. 2. The probability that the home does not have a garage attached given that it is township 5 is 0.25.
3. 3. The probability that the home has a garage attached and is in township 3 is 0.14.
4. 4. The probability that the home does not have a garage attached or is in township 2is 0.47.

Explanation of Solution

Calculation:

The below table shows the number of homes that have a garage versus the number of homes that do not have a garage in each of the five townships.

	Township
garage	1	2	3	4	5	Total
Yes	9	15	15	20	12	71
No	6	5	10	9	4	34
Total	15	20	25	29	16	105

Home has a garage attached:

The probability that the home has a garage attached is obtained as follows:

$\begin{array}{c}P\left(\text{home has a garage}\right)=\frac{\text{Number of homes that have garage}}{\text{Total number of homes}}\\ =\frac{71}{105}\\ =0.68\end{array}$

Thus, the probability that the home has a garage attached is 0.68.

Home does not have a garage attached given that it is in township 5:

The probability that the home does not have a garage attached given that it is in township 5 is obtained as follows:

$\begin{array}{c}P\left(\begin{array}{l}\text{home does not have a garage attached }\\ \text{given that it is in township 5}\end{array}\right)=\frac{\left(\begin{array}{l}\text{Number of homes that are not garage }\\ \text{attached and is in township 5 }\end{array}\right)}{\text{Total number of homes in township 5}}\\ =\frac{4}{16}\\ =0.25\end{array}$

Thus, the probability that the home does not have a garage attached given that it is in township 5 is 0.25.

Home has a garage attached and is in township 3:

The probability that the home has a garage attached and is in township 3 is obtained as follows:

$\begin{array}{c}P\left(\begin{array}{l}\text{home has a garage attached }\\ \text{and is in township 3}\end{array}\right)=\frac{\left(\begin{array}{l}\text{Number of homes that are }\\ \text{garage attached and is in township 3 }\end{array}\right)}{\text{Total number of homes}}\\ =\frac{15}{105}\\ =0.14\end{array}$

Thus, the probability that the home has a garage attached and is in township 3 is 0.14.

Home does not have a garage attached or in township 2:

The probability that the home does not have a garage attached or in township 2 is obtained as follows:

$\begin{array}{c}P\left(\begin{array}{l}\text{home does not }\\ \text{have a garage attached}\\ \text{or in township 2}\end{array}\right)=\frac{\left(\begin{array}{l}\text{Number of homes that are not garage attached +}\\ \text{Number of homes that are in township 2}-\\ \text{Number of homes that are not garage attached }\\ \text{and is in township 2 }\end{array}\right)}{\text{Total number of homes}}\\ =\frac{34+20-5}{105}\\ =\frac{49}{105}\\ =0.47\end{array}$

Thus, the probability that the home does not have a garage attached or in township 2 is 0.47.