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Chapter 4: Problem 147

Show that if event \(\mathrm{A}\) is a subset of event \(\mathrm{B}\), then \(P(\mathrm{A} \text { or } \mathbf{B})=P(\mathbf{B})\)

Short Answer

Expert verified
Given event A is a subset of event B, the sum of the probability of A and the probability of B minus the probability of A (since it is a subset of B) does prove the equation \(P(\mathrm{A} \text { or } \mathbf{B}) = P(\mathbf{B})\). This is because all outcomes of A also make B occur.

Step by step solution

01

Define the terms

An 'event' in probability theory is a set of outcomes of an experiment. A 'subset' is a set contained in another set, so if event A is a subset of event B, then every outcome that makes A occur also makes B occur. The 'probability' of an event is a measure of the likeliness that the event will occur.
02

Apply the definitions to the equation

Knowing that event A is a subset of event B, we can infer that \(P(\mathrm{A} \text { or } \mathbf{B}) = P(\mathbf{B})\) because all outcomes that would make A occur would also make B occur.
03

Analyze the equation

Generally, to find the probability of event A or B happening, we would add the probability of A happening to the probability of B happening, and subtract the probability of both A and B happening to avoid double counting. However, in this case, since A is a subset of B, there are no outcomes in which A happens but B does not. So, the probability of both A and B happening is simply the probability of A happening, which we can term as \(P(\mathbf{A})\).
04

Apply the concept to the equation and prove

So if we return to the equation and substitute the definitions, we get: \(P(\mathrm{A} \text { or } \mathbf{B}) = P(\mathbf{A}) + P(\mathbf{B}) - P(\mathbf{A})\). Essentially, we're adding the probability of A and B and subtracting A, because A happens every time B does (since A is a subset of B). Therefore, we get the result \(P(\mathrm{A} \text { or } \mathbf{B}) = P(\mathbf{B})\).

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Key Concepts

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

Subset of an Event
In probability theory, understanding the relationship between events as subsets is crucial. Whenever you have event A as a subset of event B, it means that every time event A happens, event B automatically happens as well. Think of it like this: A can be seen as a smaller part of B.
For example, consider event B to be all the rainy days in a month, and event A as all the stormy days in the same month. Every stormy day is indeed a rainy day, making A a subset of B. Understanding this helps you simplify probability calculations later on.
Probability of an Event
Probability is a fundamental concept in statistics, defined as the measure of the likelihood that a specific event will occur. This is expressed numerically between 0 and 1, where 0 indicates impossibility, and 1 indicates certainty.
The probability of an event can be calculated using the formula:
  • \( P(A) = \frac{\text{Number of favorable outcomes for A}}{\text{Total number of possible outcomes}} \)

For instance, rolling a die to get an even number has favorable outcomes 2, 4, and 6. The probability would be calculated as \( \frac{3}{6} = 0.5 \). Recognizing how to calculate probability is a key skill, especially when dealing with subsets of events.
Union of Events
The union of events, expressed as \(P(A \text{ or } B)\), represents the probability that at least one of the events A or B will occur. When you think of the union, imagine combining all outcomes from both events. However, be careful not to double-count outcomes that belong to both A and B.
In mathematics, you express the union as:
  • \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)

If A is a subset of B, you can simplify the calculation. For example, for subset events, \(P(A \cup B)\) simplifies to \(P(B)\) because all parts of A are covered in B. Understanding how to effectively use the union of events helps simplify complex probability calculations.

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Most popular questions from this chapter

Simulates generating a family. The "family" will stop having children when it has a boy or three girls, whichever comes first. Assuming that a woman is equally likely to bear a boy or a girl, perform the simulation 24 times. What is the probability that the family will have a boy?

According to the American Pet Products Manufacturers Association \(2007-2008\) National Pet Owners Survey, about \(63 \%\) of all American dog owners-some 60 million-are owners of one dog. Based on this information, find the probability that an American dog owner owns more than one dog.

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If \(P(\mathrm{A})=0.4, P(\mathrm{B})=0.5,\) and \(P(\mathrm{A} \text { and } \mathrm{B})=0.1\) find \(P(\mathrm{A} \text { or } \mathrm{B})\)
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