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 and , which is:
Where:
- is the probability of selecting an order from Restaurant A.
- is the probability of selecting an accurate order.
- 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:
Step 1: Probability of selecting an order from Restaurant A, :
The number of orders from Restaurant A is:
Step 2: Probability of selecting an accurate order, :
The number of accurate orders is:
Step 3: Probability of selecting an accurate order from Restaurant A, :
The number of accurate orders from Restaurant A is 324. So:
Step 4: Calculate the probability of getting an order from Restaurant A or an accurate order:
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:
- What is the probability of getting an order from Restaurant B or an accurate order?
- How would the probability change if the total number of orders increased?
- What is the probability of getting an inaccurate order from Restaurant C?
- What is the combined probability of selecting an accurate order from Restaurants A and D?
- 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
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