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
327327
276276
235235
120120
Order Not Accurate
4040
5858
3838
1212
If two orders are selected, find the probability that they are both from Restaurant D.
a. Assume that the selections are made with replacement. Are the events independent?
b. Assume that the selections are made without replacement. Are the events independent?
Question content area bottom
Part 1
a. Assume that the selections are made with replacement. Are the events independent?
The probability of getting two orders from Restaurant D is
0.01420.0142.
The events
are not
independent because choosing the first order
affects
the probability of the choice of the second order.
(Do not round until the final answer. Round to four decimal places as needed.)
Part 2
b. Assume that the selections are made without replacement. Are the events independent?
The probability of getting two orders from Restaurant D is
enter your response here.
The events
▼ are not
are
independent because choosing the first order
▼ affects
does not affect
the probability of the choice of the second order.
(Do not round until the final answer. Round to four decimal places as needed.)
Solution
Let's first compute the total number of orders from all restaurants and then proceed to answer the questions.
The total number of orders for each restaurant is the sum of accurate and not accurate orders:
- Restaurant A:
- Restaurant B:
- Restaurant C:
- Restaurant D:
The total number of orders across all restaurants is:
Part 1: Selections with Replacement
When the selections are made with replacement, the probability of selecting an order from Restaurant D on each draw is:
Since the selections are made with replacement, the probability of selecting two orders from Restaurant D is:
Let's compute this:
So, the probability is approximately 0.0142.
Since the selections are made with replacement, the events are independent because the outcome of the first selection does not affect the probability of the second selection.
Part 2: Selections without Replacement
When the selections are made without replacement, the probability of selecting an order from Restaurant D on the first draw is:
After selecting one order from Restaurant D, there are only 131 orders left from Restaurant D and 1105 total orders remaining. So, the probability of selecting a second order from Restaurant D is:
Thus, the probability of selecting two orders from Restaurant D without replacement is:
Let's compute this:
So, the probability is approximately 0.0141.
Since the selections are made without replacement, the events are not independent because the first selection affects the probability of the second selection.
Summary:
- Part 1 (with replacement): The probability is 0.0142 and the events are independent.
- Part 2 (without replacement): The probability is 0.0141 and the events are not independent.
Let me know if you need any further clarification or details.
Here are some related questions:
- What is the probability of getting one accurate order from Restaurant D?
- How does the probability change if we select three orders instead of two?
- What is the probability of selecting one accurate and one not accurate order from Restaurant A?
- How do probabilities compare across different restaurants for two orders?
- How would the probability change if Restaurant D had more orders?
Tip: When calculating probabilities with or without replacement, always consider whether the first event affects the second event, as this determines independence.
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Math Problem Analysis
Mathematical Concepts
Probability
Independent and Dependent Events
Probability with and without Replacement
Formulas
P(A and B) with replacement = P(A) * P(B)
P(A and B) without replacement = P(A) * P(B after A)
Theorems
Multiplication Rule for Probability
Concept of Independence in Probability
Suitable Grade Level
Grades 10-12
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