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Chapter 4: Problem 60

A college student frequents one of two coffee houses on campus, choosing Starbucks \(70 \%\) of the time and Peet's \(30 \%\) of the time. Regardless of where she goes, she buys a cafe mocha on \(60 \%\) of her visits. a. The next time she goes into a coffee house on campus, what is the probability that she goes to Starbucks and orders a cafe mocha? b. Are the two events in part a independent? Explain. c. If she goes into a coffee house and orders a cafe mocha, what is the probability that she is at Peet's? d. What is the probability that she goes to Starbucks or orders a cafe mocha or both?

Short Answer

Expert verified
Answer: The probability that the student goes to Starbucks or orders a cafe mocha or both is 0.88.

Step by step solution

01

Calculate P(Cafe Mocha | Starbucks)

Since the probability of buying a cafe mocha on any visit is 60%, we have: P(Cafe Mocha | Starbucks) = 0.60.
02

Determine P(Starbucks and Cafe Mocha) using conditional probability

We use the formula for conditional probability: P(Starbucks and Cafe Mocha) = P(Starbucks) * P(Cafe Mocha | Starbucks) = 0.70 * 0.60 = 0.42. Answer to part a: The probability that the student goes to Starbucks and orders a cafe mocha is 0.42. #b. Are the two events in part a independent?#
03

Check the independence of events

To check the independence of events, we must verify if P(Starbucks and Cafe Mocha) = P(Starbucks) * P(Cafe Mocha). We already have P(Starbucks and Cafe Mocha) = 0.42 and P(Starbucks) = 0.70. We need to calculate P(Cafe Mocha). P(Cafe Mocha) = P(Cafe Mocha | Starbucks) * P(Starbucks) + P(Cafe Mocha | Peet's) * P(Peet's) = 0.60 * 0.70 + 0.60 * 0.30 = 0.60. Now, we can check the independence of events. P(Starbucks and Cafe Mocha) = 0.42 P(Starbucks) * P(Cafe Mocha) = 0.70 * 0.60 = 0.42 Since both probabilities equal 0.42, the two events are independent. Answer to part b: The events are independent. #c. The probability of being at Peet's given that the student orders a Cafe Mocha#
04

Determine P(Peet's | Cafe Mocha) using Bayes' theorem

We use Bayes' theorem for conditional probability: P(Peet's | Cafe Mocha) = P(Cafe Mocha | Peet's) * P(Peet's) / P(Cafe Mocha) = (0.60 * 0.30) / 0.60 = 0.30. Answer to part c: The probability that the student is at Peet's given she orders a cafe mocha is 0.30. #d. The probability of going to Starbucks or ordering a cafe mocha or both#
05

Calculate P(Starbucks or Cafe Mocha or both) using the formula

We use the formula for the probability of either event A or event B or both occurring: P(Starbucks or Cafe Mocha or both) = P(Starbucks) + P(Cafe Mocha) - P(Starbucks and Cafe Mocha) = 0.70 + 0.60 - 0.42 = 0.88. Answer to part d: The probability that the student goes to Starbucks or orders a cafe mocha or both is 0.88.

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