91Ó°ÊÓ

A family consisting of three people- \(\mathrm{P}_{1}, \mathrm{P}_{2}\), and \(\mathrm{P}_{3}\) -belongs to a medical clinic that always has a physician at each of stations 1,2, and \(3 .\) During a certain week, each member of the family visits the clinic exactly once and is randomly assigned to a station. One experimental outcome is \((1,2,1)\), which means that \(\mathrm{P}_{1}\) is assigned to station \(1, \mathrm{P}_{2}\) to station 2, and \(\mathrm{P}_{3}\) to station \(1 .\) a. List the 27 possible outcomes. (Hint: First list the nine outcomes in which \(\mathrm{P}_{1}\) goes to station 1, then the nine in which \(\mathrm{P}_{1}\) goes to station 2, and finally the nine in which \(\mathrm{P}_{1}\) goes to station 3 ; a tree diagram might help.) b. List all outcomes in the event \(A\), that all three people go to the same station. c. List all outcomes in the event \(B\), that all three people go to different stations. d. List all outcomes in the event \(C\), that no one goes to station 2 . e. Identify outcomes in each of the following events: \(B^{C}, C^{C}, A \cup B, A \cap B, A \cap C\).

Step by step solution

01

Identify Possible Outcomes

First, list down all possible combinations where each person can go to one of the three stations. This should give 27 outcomes as there are 3 options (stations) for each of the 3 people (\(3^3 = 27\)).

02

Identify Outcomes for Event A

Next, list down the outcomes where all three people go to the same station. This would be the scenarios where all three people go to station 1, or all go to station 2, or all go to station 3. This should result in 3 outcomes.

03

Identify Outcomes for Event B

Now list down all outcomes where each person goes to a different station. This would be the scenarios where each of the three stations has one and only one person. There should be 6 such outcomes.

04

Identify Outcomes for Event C

Next, list down the outcomes where none of the people go to station 2. This means all the people should either go to station 1 or station 3. This should result in 9 outcomes.

05

Identify Outcomes for other events

Finally, identify the outcomes for each of the following: \(B^C\) (outcomes that are in B and not in C), \(C^C\) (outcomes that are not in C), \(A \cup B\) (outcomes that are in A or B or both), \(A \cap B\) (outcomes that are in both A and B), \(A \cap C\) (outcomes that are in both A and C).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Experimental Outcomes

Understanding experimental outcomes is crucial for grasping basic probability. In probability theory, an experimental outcome is one possible result of an experiment. For example, when you flip a coin, there are two possible experimental outcomes: heads or tails. In the context of our exercise, the experiment involves assigning each of three family members to one of the three clinic stations.

An experimental outcome would thus detail the specific station each family member goes to, represented as a triplet like \(1, 2, 1\). There are \((3 \times 3 \times 3 = 27)\) such outcomes because there are three independent choices being made. When trying to list all possible outcomes, it's helpful to adopt a systematic approach. Begin with one variable, say \(\mathrm{P}_{1}\), and list all possibilities for them before moving on to the next family member. This way, you'll ensure you've covered every combination without repetition or omission.

Event Probability

Event probability quantifies the likelihood of a specific outcome or set of outcomes occurring. It’s calculated by dividing the number of ways a particular event can occur by the total number of possible outcomes. The probability of event \(A\), for example, is calculated by counting the number of outcomes where all three family members visit the same station—there are 3 such outcomes—and dividing by the total number of experimental outcomes, which is 27.

So, the probability of event \(A\) happening is \(\frac{3}{27} = \frac{1}{9}\). Remember, the sum of the probabilities of all possible experimental outcomes must equal 1, which represents certainty. This concept helps us predict the chance of various scenarios and is fundamental to many real-world applications, from weather forecasting to game theory.

Tree Diagram

A tree diagram is a graphical representation that helps in visualizing all possible outcomes of an experiment, and it's particularly useful for understanding compound events. Imagine a branching tree: starting from a single point (the 'root'), it splits into branches that represent all possible outcomes of a first event. Each of these branches then splits further to represent the outcomes of a second event, and so on.

In our clinic example, a tree diagram would start with three branches for \(\mathrm{P}_{1}\), each leading to three further branches for \(\mathrm{P}_{2}\), and each of those splitting into three more for \(\mathrm{P}_{3}\). By the end of the diagram, you'd have 27 distinct 'leaves' (endpoints), representing each experimental outcome. Tree diagrams provide an intuitive way to count all possible outcomes and can be very useful in calculating probabilities.

Compound Events

A compound event in probability consists of two or more simple events happening simultaneously. For instance, events like \(A\), \(B\), and \(C\) from our exercise involve multiple individuals ending up at certain stations, making them compound events. To derive probabilities of compound events, one often combines probabilities of simpler events using addition or multiplication rules.

In our scenario, event \(A\) (all members at the same station) and event \(B\) (all members at different stations) are particularly interesting. Analyzing compound events often requires understanding the concepts of 'union' and 'intersection'. For instance, \(A \cap B\) represents the intersection of events \(A\) and \(B\), and would occur only if both \(A\) and \(B\) were true for an outcome. However, since \(A\) and \(B\) are mutually exclusive (cannot both be true simultaneously), their intersection, \(A \cap B\), would not contain any outcomes. Such insights are crucial when assessing complex scenarios.