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Chapter 3: Problem 9

Disjoint vs. independent. In parts (a) and (b), identify whether the events are disjoint, independent, or neither (events cannot be both disjoint and independent). (a) You and a randomly selected student from your class both earn A's in this course. (b) You and your class study partner both earn A's in this course. (c) If two events can occur at the same time, must they be dependent?

Short Answer

Expert verified
(a) Independent, (b) Neither, (c) No, they need not be dependent.

Step by step solution

01

Understanding Disjoint Events

Disjoint events, also known as mutually exclusive events, are those that cannot happen at the same time. If Event A occurs, Event B cannot occur, and vice versa. Mathematically, for two events A and B to be disjoint, the intersection of A and B must be empty, i.e., \( P(A \cap B) = 0 \).
02

Understanding Independent Events

Independent events are events where the occurrence of one event does not affect the probability of the other event occurring. Mathematically, two events A and B are independent if and only if \( P(A \cap B) = P(A) \cdot P(B) \).
03

Analyzing Part (a)

In part (a), 'You and a randomly selected student both earn A's'. These are not disjoint since both can occur simultaneously (both you and the student can get A's). We need to analyze independence. Typically, two students' grades are independent unless tied by some shared factor (e.g., sharing answers), so they can be considered independent. Thus, the events are independent.
04

Analyzing Part (b)

In part (b), 'You and your class study partner both earn A's'. These events are not disjoint for similar reasons as (a). However, they are likely not independent because studying together could influence both parties' performances. Thus, the events in this scenario are neither disjoint nor independent (they might be dependent, but not disjoint).
05

Analyzing Part (c)

For part (c), the question asks if two events that can occur at the same time must be dependent. The answer is no, as exemplified by independent events that can occur simultaneously without affecting each other's probabilities (e.g., flipping two different coins and getting heads on both). Thus, they can be independent.

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Key Concepts

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

Disjoint Events
Imagine two events that cannot happen at the same time. This is what we call disjoint events, and they're easy to identify once you know this key trait: if one event happens, the other must not. A clear example is rolling a die. If you roll a 3, you cannot also roll a 4 simultaneously. Mathematically, disjoint events have no outcomes in common, which means the probability of both events occurring is zero: \( P(A \cap B) = 0 \). If you're ever asked to identify disjoint events, simply check if both can happen at the same time. If not, they're disjoint!
Keep in mind that disjoint events are also called mutually exclusive events, so be ready for both terms to pop up in your studies.
Independent Events
Let's dive into the world of independent events. These are events where the outcome of one does not influence the other. A classic example involves flipping two different coins. The result of one coin flip (say, heads) has no effect on the result of the other coin flip. We express this mathematically by showing that the probability of both independent events occurring together is the product of their individual probabilities: \( P(A \cap B) = P(A) \cdot P(B) \).
So, when tasked with identifying independent events, ask yourself if knowing the outcome of one event changes the likelihood of the other. If the answer is no, then you have found independent events.
Mutually Exclusive Events
Mutually exclusive events are essentially another term for disjoint events. These events cannot happen at the same time. If Event A occurs, then Event B cannot, and vice versa. It's important to remember their defining feature: no overlap. This means, mathematically, \( P(A \cap B) = 0 \). Consider drawing a card from a deck; landing both a heart and a club on the same card is impossible, hence it's mutually exclusive.
In practical examples, whenever you see events that are labeled as mutually exclusive, check if their happening has any overlap. If there's none, then you are dealing with mutually exclusive or disjoint events.
Dependent Events
Dependent events occur when the outcome or occurrence of one event affects the probability of another. Unlike independent events, here, what happens with one event provides some sort of information about the other. For instance, consider drawing two cards from a deck without replacement. The value of the first card will influence the probability of drawing a certain kind of card second. The mathematical expression of dependency is shown through conditional probability.
If you're trying to determine whether events are dependent, ask yourself if the result of one event informs the probability of the second. If so, these events are likely dependent.

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