Questions: Let's eat: A fast-food restaurant chain has 604 outlets in the United States. The following table categorizes them by city population size and location, and presents the number of restaurants in each category. A restaurant is to be chosen at random from the 604 to test market a new menu. Round your answers to four decimal places. Population of City Region --- NE SE SW NW Under 50,000 - - - - 50,000-500,000 34 40 13 9 Over 500,000 56 88 70 33 (a) Given that the restaurant is located in a city with a population over 500,000, what is the probability that it is in the Northeast? (b) Given that the restaurant is located in the Southeast, what is the probability that it is in a city with a population under 50,000? (c) Given that the restaurant is located in the Southwest, what is the probability that it is in a city with a population of 500,000 or less?

Let's eat: A fast-food restaurant chain has 604 outlets in the United States. The following table categorizes them by city population size and location, and presents the number of restaurants in each category. A restaurant is to be chosen at random from the 604 to test market a new menu. Round your answers to four decimal places. Population of City Region --- NE SE SW NW Under 50,000 - - - - 50,000-500,000 34 40 13 9 Over 500,000 56 88 70 33 (a) Given that the restaurant is located in a city with a population over 500,000, what is the probability that it is in the Northeast? (b) Given that the restaurant is located in the Southeast, what is the probability that it is in a city with a population under 50,000? (c) Given that the restaurant is located in the Southwest, what is the probability that it is in a city with a population of 500,000 or less?
Transcript text: Let's eat: A fast-food restaurant chain has 604 outlets in the United States. The following table categorizes them by city population size and location, and presents the number of restaurants in each category. A restaurant is to be chosen at random from the 604 to test market a new menu. Round your answers to four decimal places. \begin{tabular}{lllll} \begin{tabular}{l} Population \\ of City \end{tabular} & \multicolumn{4}{c}{ Region } \\ \hline Under 50,000 & NE & SE & SW & NW \\ \hline $50,000-500,000$ & 34 & 40 & 13 & 9 \\ Over 500,000 & 56 & 88 & 70 & 33 \\ \hline \end{tabular} (a) Given that the restaurant is located in a city with a population over 500,000, what is the probability that it is in the Northeast? (b) Given that the restaurant is located in the Southeast, what is the probability that it is in a city with a population under 50,000 ? (c) Given that the restaurant is located in the Southwest, what is the probability that it is in a city with a population of 500,000 or less?
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Solution

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Solution Steps

Solution Approach

(a) To find the probability that a restaurant is in the Northeast given that it is located in a city with a population over 500,000, use the formula for conditional probability: \( P(A|B) = \frac{P(A \cap B)}{P(B)} \). Here, \( A \) is the event that the restaurant is in the Northeast, and \( B \) is the event that the restaurant is in a city with a population over 500,000.

(b) To find the probability that a restaurant is in a city with a population under 50,000 given that it is located in the Southeast, use the same conditional probability formula. Here, \( A \) is the event that the restaurant is in a city with a population under 50,000, and \( B \) is the event that the restaurant is in the Southeast.

(c) To find the probability that a restaurant is in a city with a population of 500,000 or less given that it is located in the Southwest, use the conditional probability formula. Here, \( A \) is the event that the restaurant is in a city with a population of 500,000 or less, and \( B \) is the event that the restaurant is in the Southwest.

Step 1: Calculate Probability (NE | over 500k)

To find the probability that a restaurant is in the Northeast given that it is located in a city with a population over 500,000, we use the formula for conditional probability:

\[ P(\text{NE} | \text{over } 500k) = \frac{P(\text{NE} \cap \text{over } 500k)}{P(\text{over } 500k)} \]

From the data, we have:

  • \( P(\text{NE} \cap \text{over } 500k) = 56 \)
  • \( P(\text{over } 500k) = 247 \)

Thus,

\[ P(\text{NE} | \text{over } 500k) = \frac{56}{247} \approx 0.2267 \]

Step 2: Calculate Probability (under 50k | SE)

Next, we calculate the probability that a restaurant is in a city with a population under 50,000 given that it is located in the Southeast:

\[ P(\text{under } 50k | \text{SE}) = \frac{P(\text{under } 50k \cap \text{SE})}{P(\text{SE})} \]

From the data, we have:

  • \( P(\text{under } 50k \cap \text{SE}) = 0 \)
  • \( P(\text{SE}) = 128 \)

Thus,

\[ P(\text{under } 50k | \text{SE}) = \frac{0}{128} = 0.0 \]

Step 3: Calculate Probability (500k or less | SW)

Finally, we calculate the probability that a restaurant is in a city with a population of 500,000 or less given that it is located in the Southwest:

\[ P(\text{500k or less} | \text{SW}) = \frac{P(\text{500k or less} \cap \text{SW})}{P(\text{SW})} \]

From the data, we have:

  • \( P(\text{500k or less} \cap \text{SW}) = 13 \)
  • \( P(\text{SW}) = 83 \)

Thus,

\[ P(\text{500k or less} | \text{SW}) = \frac{13}{83} \approx 0.1566 \]

Final Answer

The probabilities are as follows:

  • (a) \( P(\text{NE} | \text{over } 500k) \approx 0.2267 \)
  • (b) \( P(\text{under } 50k | \text{SE}) = 0.0000 \)
  • (c) \( P(\text{500k or less} | \text{SW}) \approx 0.1566 \)

Thus, the final answers are: \[ \boxed{(a) \, 0.2267, \, (b) \, 0.0000, \, (c) \, 0.1566} \]

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