Math Problem Statement

Use the data in the following​ table, which lists​ drive-thru order accuracy at popular fast food chains. Assume that orders are randomly selected from those included in the table. ​Drive-thru Restaurant A B C D Order Accurate 324 273 232 148 Order Not Accurate 37 50 32 18 If one order is​ selected, find the probability of getting an order from Restaurant A or an order that is accurate. Are the events of selecting an order from Restaurant A and selecting an accurate order disjoint​ events? Question content area bottom Part 1 The probability of getting an order from Restaurant A or an order that is accurate is

enter your response here. ​(Round to three decimal places as​ needed.) Part 2 Are the events of selecting an order from Restaurant A and selecting an accurate order disjoint​ events? The events ▼ are not are disjoint because it ▼ is not is possible to ▼ pick an inaccurate order. pick an order from Restaurant B, C, or D. receive an accurate order from Restaurant A.

Solution

Part 1: Probability of getting an order from Restaurant A or an order that is accurate

To solve this, we need to use the formula for the probability of the union of two events AA and BB, which is:

P(A or B)=P(A)+P(B)P(A and B)P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

Where:

  • P(A)P(A) is the probability of selecting an order from Restaurant A.
  • P(B)P(B) is the probability of selecting an accurate order.
  • P(A and B)P(A \text{ and } B) is the probability of selecting an accurate order from Restaurant A (i.e., the intersection of the two events).

First, let's calculate the total number of orders:

Total Orders=(324+273+232+148)+(37+50+32+18)=977\text{Total Orders} = (324 + 273 + 232 + 148) + (37 + 50 + 32 + 18) = 977

Step 1: Probability of selecting an order from Restaurant A, P(A)P(A):

The number of orders from Restaurant A is:

P(A)=Orders from ATotal Orders=324+37977=3619770.369P(A) = \frac{\text{Orders from A}}{\text{Total Orders}} = \frac{324 + 37}{977} = \frac{361}{977} \approx 0.369

Step 2: Probability of selecting an accurate order, P(B)P(B):

The number of accurate orders is:

P(B)=Total Accurate OrdersTotal Orders=324+273+232+148977=977977=0.977P(B) = \frac{\text{Total Accurate Orders}}{\text{Total Orders}} = \frac{324 + 273 + 232 + 148}{977} = \frac{977}{977} = 0.977

Step 3: Probability of selecting an accurate order from Restaurant A, P(A and B)P(A \text{ and } B):

The number of accurate orders from Restaurant A is 324. So:

P(A and B)=3249770.332P(A \text{ and } B) = \frac{324}{977} \approx 0.332

Step 4: Calculate the probability of getting an order from Restaurant A or an accurate order:

P(A or B)=P(A)+P(B)P(A and B)=0.369+0.9770.332=1.014P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) = 0.369 + 0.977 - 0.332 = 1.014

Since probabilities cannot exceed 1, this result indicates that the total probability is essentially 1, meaning that getting an order from Restaurant A or an accurate order is highly likely.

Part 2: Are the events disjoint?

Two events are disjoint if they cannot happen at the same time. In this case, it is possible to select an order from Restaurant A and for that order to be accurate (i.e., you can have both events occur simultaneously).

Thus, the events are not disjoint because it is possible to receive an accurate order from Restaurant A.


Summary:

  • The probability of getting an order from Restaurant A or an order that is accurate is approximately 1 (rounded to three decimal places).
  • The events are not disjoint because it is possible to receive an accurate order from Restaurant A.

Would you like further details or have any other questions? Here are some related questions for further exploration:

  1. What is the probability of getting an order from Restaurant B or an accurate order?
  2. How would the probability change if the total number of orders increased?
  3. What is the probability of getting an inaccurate order from Restaurant C?
  4. What is the combined probability of selecting an accurate order from Restaurants A and D?
  5. If you randomly select two orders, what is the probability that both are accurate?

Tip: When dealing with "or" probabilities, remember to subtract the overlap (intersection) to avoid double-counting!

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Math Problem Analysis

Mathematical Concepts

Probability
Set Theory

Formulas

P(A or B) = P(A) + P(B) - P(A and B)

Theorems

Law of Total Probability
Union of Events in Probability

Suitable Grade Level

Grades 10-12