Chapter 4.3, Problem 39E

a.

To determine

Compute the probability that selected restaurant is located in Northeast given that the city has population over 500,000.

Answer to Problem 39E

The probability that selected restaurant is located in Northeast given that the city has population over 500,000 is 0.5660.

Explanation of Solution

Calculation:

There were a total of 600. The restaurants are classified based on city population and location. The table provides an idea about number of restaurants in each category.

The formula for conditional probability is given by:

$P\left(B|A\right)=\frac{\text{Number of out comes corresponding}\left(A\text{and}B\right)}{\text{Number of out comes corresponding}A}$.

Event A denotes that selected restaurant is located in a city with population over 500,000 and event B denotes that selected restaurant is located in Northeast. Event A contains those restaurant which are located in northeast, southeast, and southwest or in northwest region with population over 500,000.

From the data it is clear that 150 of the restaurants located in Northeast and the city has population over 500,000.

$\begin{array}{c}\text{Number of restaurants under the category over 500,000}=150+25+30+60\\ =265\end{array}$

Substitute these values in the formula for conditional probability.

Therefore,

$\begin{array}{c}P\left(B|A\right)=\frac{150}{265}\\ =0.5660\end{array}$

Thus, the probability that selected restaurant is located in Northeast given that the city has population over 500,000 is 0.5660.

b.

To determine

Compute the probability that selected restaurant is located in city with population under 500,000 given that it is located in southeast.

Answer to Problem 39E

The probability that selected restaurant is located in city with population under 50,000 given that it is located in southeast is 0.2333.

Explanation of Solution

Calculation:

The formula for conditional probability is given by:

$P\left(D|C\right)=\frac{\text{Number of out comes corresponding}\left(C\text{and }D\right)}{\text{Number of out comes corresponding}C}$.

Event C denotes that selected restaurant is located in southeast and event D denotes that selected restaurant is located in a city with population under 50,000. Event C contains those restaurants which are located in southeast region with population under 50,000; 50,000-500,000; or over 500,000.

From the data it is clear that 35 of the restaurants located in southeast and the city has population under 50,000.

$\begin{array}{c}\text{Number of restaurants in southeast}=35+90+25\\ =150\end{array}$

Substitute these values in the formula for conditional probability.

Therefore,

$\begin{array}{c}P\left(D|C\right)=\frac{35}{150}\\ =0.2333\end{array}$

Thus, the probability that selected restaurant is located in city with population under 50,000 given that it is located in southeast is 0.2333.

c.

To determine

Compute the probability that selected restaurant is located in city with population 500,000 or less given that it is located in southwest.

Answer to Problem 39E

The probability that selected restaurant is located in city with population 500,000 or less given that it is located in southwest is 0.7391.

Explanation of Solution

Calculation:

The formula for conditional probability of any two events E and F, is given by:

$P\left(F|E\right)=\frac{\text{Number of out come corresponding}\left(E\text{and }F\right)}{\text{Number of out come corresponding}E}$.

Event E denotes that selected restaurant is located in southwest and event F denotes that selected restaurant is located in a city with population 500,000 or less. Event E contains those restaurants which are located in southwest region with population under 50,000; 50,000-500,000; or over 500,000.

From the data it is clear that 15 of the restaurants located in southwest and the city has population under 50,000, 70 of the restaurants located in southwest and the city has population between 50,000-500,000.

$\begin{array}{c}\text{Number of restaurants in southwest with population less than 500,000}=15+70\\ =85\end{array}$

$\begin{array}{c}\text{Number of restaurants in southwest}=15+70+30\\ =115\end{array}$

Substitute these values in the formula for conditional probability.

Therefore,

$\begin{array}{c}P\left(F|E\right)=\frac{85}{115}\\ =0.7391\end{array}$

Thus, the probability that selected restaurant is located in city with population 500,000 or less given that it is located in southwest is 0.7391.

d.

To determine

Compute the probability that selected restaurant is located in southwest given that it is a city with population 500,000 or less.

Answer to Problem 39E

The probability that selected restaurant is located in southwest given that it is a city with population 500,000 or less is 0.2537.

Explanation of Solution

Calculation:

The formula for conditional probability of any two events E and F, is given by:

$P\left(E|F\right)=\frac{\text{Number of out come corresponding}\left(E\text{and }F\right)}{\text{Number of out come corresponding}F}$.

Event E denotes that selected restaurant is located in southwest and event F denotes that selected restaurant is located in a city with population 500,000 or less. Event F contains those restaurants which are located in any of the four regions with population under 50,000 or restaurants which are located in any of the four regions with population 50,000-500,000.

From part (c). it is clear that 85 of the restaurants located in southwest has population 500,000 or less.

$\begin{array}{c}\text{Number of restaurants in city with population 500,000 or less }=\left[\begin{array}{l}30+35+15+5+60+\\ 90+70+30\end{array}\right]\\ =335\end{array}$

Substitute these values in the formula for conditional probability.

Therefore,

$\begin{array}{c}P\left(E|F\right)=\frac{85}{335}\\ =0.2537\end{array}$

Thus, the probability that selected restaurant is located in southwest given that it is a city with population 500,000 or less is 0.2537.

e.

To determine

Compute the probability that selected restaurant is located in city with population 50,000 or more given that it is located in south.

Answer to Problem 39E

The probability that selected restaurant is located in city with population 500,000 or more given that it is located in south is 0.8113.

Explanation of Solution

Calculation:

The formula for conditional probability of any two events G and H is given by:

$P\left(H|G\right)=\frac{\text{Number of out come corresponding}\left(G\text{and }H\right)}{\text{Number of out come corresponding}G}$.

Event G denotes that selected restaurant is located in south and event H denotes that selected restaurant is located in a city with population 50,000 or more. Event G contains those restaurants which are located in southeast and any of the three population size or restaurants which are located in southwest and any of the three population size.

From the data it is clear that $25+90=115$ of the restaurants located in southeast and the city has population 50,000 or more, $30+70=100$ of the restaurants located in southwest and the city has population 50,000 or more.

$\begin{array}{c}\text{Number of restaurants in south}\text{with population 50,000}\text{or more}=115+100\\ =215\end{array}$

$\begin{array}{c}\text{Number of restaurants in south}=\left[35+90+25+15+70+30\right]\\ =265\end{array}$

Substitute these values in the formula for conditional probability.

Therefore,

$\begin{array}{c}P\left(H|G\right)=\frac{215}{265}\\ =0.8113\end{array}$

Thus, the probability that selected restaurant is located in city with population 500,000 or more s given that it is located in south is 0.8113.