91Ó°ÊÓ

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 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.

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 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.