Question

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

Answer: The probability that she goes to Starbucks or orders a cafe mocha or both is 0.88.

Step-by-step solution

Identify the probabilities

We are given the following probabilities: - The probability of going to Starbucks, Pr(S), is 0.7. - The probability of going to Peet's, Pr(P), is 0.3. - The probability of ordering a cafe mocha in either coffee house, Pr(M), is 0.6.

Calculate joint probability

To find the probability that she goes to Starbucks and orders a cafe mocha, we need to calculate the joint probability of these events. Using the multiplication rule: Pr(S and M) = Pr (S) * Pr(M|S) Since Pr(M) is given for both coffee houses, we can consider this as Pr(M|S) for Starbucks. Thus, Pr(S and M) = Pr(S) * Pr(M|S) = 0.7 * 0.6 = 0.42 So, the probability that she goes to Starbucks and orders a cafe mocha is 0.42. #b. Are the two events in part a independent? Explain.#

Examine the joint probability

To determine if the events S (going to Starbucks) and M (ordering a cafe mocha) are independent, we need to check if the joint probability Pr(S and M) equals the product of their individual probabilities: If Pr(S and M) = Pr(S) * Pr(M), then the events are considered independent.

Check for independence

We already calculated the joint probability Pr(S and M) as 0.42. Now we need to check if Pr(S) * Pr(M) equals 0.42: Pr(S) * Pr(M) = 0.7 * 0.6 = 0.42 Since Pr(S and M) = Pr(S) * Pr(M), the events are independent. #c. If she goes into a coffee house and orders a cafe mocha, what is the probability that she is at Peet's?#

Calculate the joint probability of P and M

To find the probability that she goes to Peet's and orders a cafe mocha, we need to calculate the joint probability of these events, using the multiplication rule: Pr(P and M) = Pr(P) * Pr(M|P) Since Pr(M) is given for both coffee houses, we can consider this as Pr(M|P) for Peet's. Thus, Pr(P and M) = Pr(P) * Pr(M|P) = 0.3 * 0.6 = 0.18 So, the probability that she goes to Peet's and orders a cafe mocha is 0.18.

Calculate the conditional probability that she is at Peet's after ordering a cafe mocha

To find the probability that she is at Peet's after ordering a cafe mocha, we will use the conditional probability formula: Pr(P|M) = Pr(P and M) / Pr(M) We already calculated Pr(P and M) as 0.18 in step 1. To find Pr(M): Pr(M) = Pr(S and M) + Pr(P and M) = 0.42 + 0.18 = 0.6 Now, we calculate the conditional probability: Pr(P|M) = Pr(P and M) / Pr(M) = 0.18 / 0.6 = 0.3 So, the probability that she is at Peet's after ordering a cafe mocha is 0.3. #d. What is the probability that she goes to Starbucks or orders a cafe mocha or both?#

Use the addition rule of probability

To find the probability that a student goes to Starbucks or orders a cafe mocha or both, we can use the addition rule of probability: Pr(S or M) = Pr(S) + Pr(M) - Pr(S and M) We already calculated the following probabilities: Pr(S) = 0.7, Pr(M) = 0.6, and Pr(S and M) = 0.42.

Calculate the probability

Now, we can calculate Pr(S or M) = Pr(S) + Pr(M) - Pr(S and M) = 0.7 + 0.6 - 0.42 = 0.88 The probability that she goes to Starbucks or orders a cafe mocha or both is 0.88.

Key concepts

Independent Events

In probability, independent events are those where the occurrence of one event does not affect the occurrence of the other. A simple example can be rolling two dice. The outcome of one die doesn't influence what happens with the other.
In our exercise, we have two events: going to Starbucks (S) and ordering a cafe mocha (M). We found that these events are independent by calculating their joint probability and comparing it with the product of their separate probabilities.
The joint probability was calculated as \(Pr(S \text{ and } M) = 0.42\), while the product of their individual probabilities is \(Pr(S) \cdot Pr(M) = 0.42\). Since these values are equal, it confirms that going to Starbucks and ordering a cafe mocha are independent events.
This means the student's decision to visit Starbucks doesn't impact her choice to order a cafe mocha.

Conditional Probability

Conditional probability is used to find the probability of an event occurring given that another event has already occurred. In simple terms, it answers the question: "What's the chance of event A happening if we know that event B happened?"
For our scenario, we calculated the conditional probability of the student being at Peet's, knowing she ordered a cafe mocha. This was expressed as \(Pr(P|M)\), which is calculated using the formula:

• \(Pr(P|M) = \frac{Pr(P \text{ and } M)}{Pr(M)}\)

We found \(Pr(P \text{ and } M) = 0.18\) and \(Pr(M) = 0.6\). Substituting these values, \(Pr(P|M) = \frac{0.18}{0.6} = 0.3\).
This means that if she orders a cafe mocha, there's a 30% probability she's at Peet's.

Addition Rule of Probability

The addition rule of probability helps determine the probability of either of two events occurring. It is especially helpful when dealing with events that can occur simultaneously.
According to the addition rule:\[Pr(A \text{ or } B) = Pr(A) + Pr(B) - Pr(A \text{ and } B)\]
This formula accounts for any overlap between the events. In our example, we calculated the probability that the student goes to Starbucks or orders a cafe mocha or both. The values used were:

• \(Pr(S) = 0.7\)
• \(Pr(M) = 0.6\)
• \(Pr(S \text{ and } M) = 0.42\)

This results in \(Pr(S \text{ or } M) = 0.7 + 0.6 - 0.42 = 0.88\).
The interpretation here is that there is an 88% chance that she will either visit Starbucks, order a cafe mocha, or do both.

Joint Probability

Joint probability refers to the likelihood of two events happening at the same time. It's useful for understanding how two events are related.
In our exercise, joint probability was used to find the probability that a student goes to Starbucks and orders a cafe mocha. We calculated this using the multiplication rule of probability:

• \(Pr(S \text{ and } M) = Pr(S) \cdot Pr(M|S)\)

Given \(Pr(S) = 0.7\) and \(Pr(M|S) = 0.6\), it was calculated to be \(0.42\).
This means that there is a 42% chance that when the student visits a coffee house, she'll both go to Starbucks and order a cafe mocha.