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Chapter 11: Problem 75

Explain the difference between mutually exclusive events and complementary events.

Short Answer

Expert verified
Mutually exclusive events cannot occur simultaneously, while complementary events are opposites and cover all possible outcomes.

Step by step solution

01

Define mutually exclusive events

Mutually exclusive events are two or more events that cannot occur simultaneously. If one event happens, the other(s) cannot. For example, when flipping a coin, getting heads and tails are mutually exclusive events because you cannot get both heads and tails at the same time.
02

Define complementary events

Complementary events are a pair of events where one event occurs if and only if the other does not. When one happens, it completely rules out the other. For example, in a coin flip, getting heads and not getting heads (i.e., getting tails) are complementary events because one is the exact opposite of the other.
03

Compare the two concepts

Mutually exclusive events are simply events that do not intersect, meaning they cannot happen at the same time. Complementary events, on the other hand, not only cannot happen at the same time, but also cover all possible outcomes of an experiment. When you add the probabilities of complementary events together, you always get 1.
04

Provide examples

For mutually exclusive events: Rolling a die and getting a 3 or a 5. These cannot occur at the same time. For complementary events: Rolling a die and getting a number that is either 3 or not 3. These events are complementary because one is the opposite of the other.

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Key Concepts

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

Mutually Exclusive Events
To understand mutually exclusive events, think of events that cannot happen at the same time. Imagine you're flipping a coin.
If you get heads, you cannot get tails on the same flip.
These outcomes are mutually exclusive.
Another example is rolling a single die.
You can't roll a 3 and a 5 at the same time.
This means that events getting a 3 and getting a 5 are mutually exclusive.
In probability terms, for any two mutually exclusive events A and B, \(P(A \text{ and } B) = 0\).
This equation means the probability of both events occurring together is zero because it's impossible for both of them to happen simultaneously.
Complementary Events
Complementary events are pairs of events where one event occurs if and only if the other does not.
Taking the coin flip example again, getting heads and not getting heads (getting tails) are complementary events.
They cover all possible outcomes of the coin flip.
If you flip a coin, you must get either heads or tails, and never both.
The sum of the probabilities of complementary events is always 1.
So, if A and B are complementary, \(P(A) + P(B) = 1\).
For a die roll, getting a 3 and not getting a 3 are complementary.
One of these must happen on any roll, covering all possible outcomes.
Probability
Probability measures the likelihood of an event happening.
It is represented as a number between 0 and 1.
If an event has a probability of 0, it means it's impossible.
A probability of 1 means the event is certain to happen.
A fair coin flip has a probability of 0.5 (or 50%) for landing heads, and 0.5 for tails.
In general, for any event A, \(P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\).
For example, rolling a 3 on a six-sided die has a probability of \(\frac{1}{6}\).
Understanding probability helps solve problems involving mutually exclusive and complementary events, by applying the correct formulas and principles.

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