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Chapter 1: Problem 16

Use Exercise 15 to show that \(P(E \cup F)=P(E)+P(F)-P(E F)\).

Short Answer

Expert verified
In order to show that \(P(E \cup F) = P(E) + P(F) - P(EF)\), we used a Venn diagram to visualize the events E, F, and their union and intersection. The sum of probabilities of E and F contains the probability of their intersection twice, so we need to subtract it once to get the correct probability of the union: \(P(E \cup F) = P(E) + P(F) - P(EF)\). The derivation is based on the basic properties of probabilities and the definition of the union and intersection of events.

Step by step solution

01

Understand and define E, F, and their union and intersection

Before we start the proof, let's understand and define the events, their union, and intersection. - Event E: An event that occurs in a probability space. - Event F: Another event that occurs in the same probability space. - Union (\(E \cup F\)): This event occurs if either E or F or both occur. - Intersection (\(EF\)): This event occurs if both E and F occur together.
02

Recall the properties of probabilities

Now, let's recall some basic properties of probabilities: 1. \(0 \leq P(E) \leq 1\): Probability of an event always lies between 0 and 1. 2. \(P(E') = 1 - P(E)\): Probability of the complement of E is equal to one minus the probability of E. 3. \(P(A \cup B) = P(A) + P(B) - P(AB)\) for any two events A and B.
03

Derive the formula using Venn diagram and properties of probabilities

We will now use a Venn diagram to help us visualize and derive the formula for the probability of the union of two events. We can draw two overlapping circles, one for event E and the other for event F. There are three distinct regions in this diagram: - the region where only E occurs; - the region where only F occurs; - the region where both E and F occur (their intersection). The probability of the union of events E and F (the area covered by both circles) is given by the sum of the probabilities of each region: \(P(E \cup F) = P(E) + P(F) - P(EF)\). To show this, we can use the properties of probabilities mentioned in step 2: 1. Add the probabilities of E and F: \(P(E) + P(F)\). This sum includes the probability of their intersection twice because both E and F contain the region where they overlap. 2. Therefore, we have to subtract the intersection probability once: \(P(E) + P(F) - P(EF)\). 3. This gives us the final formula for the probability of the union of events E and F: \(P(E \cup F) = P(E) + P(F) - P(EF)\). As we have derived the formula using the Venn diagram and properties of probabilities, we have shown that \(P(E \cup F) = P(E) + P(F) - P(EF)\).

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Key Concepts

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

Union of Events
When dealing with probability theory, understanding the concept of the union of events is crucial. The union of two events, denoted as \(E \cup F\), represents all outcomes that belong to either event E, event F, or both. Think of it as a combination of the possibilities from both events.

In more relatable terms, if E is the event of rolling a die and getting a number less than 4, and F is the event of getting an even number, \(E \cup F\) encompasses actions like rolling a 1, 2, 3, 4, or 6. Essentially, the union covers all scenarios each individual event could provide.

To find the probability of \(E \cup F\), you add the probabilities of E and F, but you need to subtract the probability of both events occurring simultaneously. This subtraction is crucial because if you simply add the probabilities, the overlap (where both E and F occur) is counted twice.
Venn Diagram
Venn Diagrams are powerful tools used to visualize relationships between different sets or events. In the context of probability theory, they offer a clear visual representation of how different events like E and F overlap or remain distinct.

Imagine two overlapping circles; one represents the events in E, and the other those in F. The area where the circles overlap is the intersection of the two events. Each part of the diagram helps to visually separate where outcomes include just E, just F, or both (the intersection). This can make it easier to see why the formula for the union of two events includes that subtraction of the intersection.

Using a Venn Diagram for the probability of \(E \cup F\), you visualize each segment's contribution to the whole. This visualization aids in understanding why we subtract \(P(EF)\). It's about carving out the double-counted area of intersection, giving you the correct probability for the union.
Intersection of Events
The intersection of two events, denoted as \(EF\) or sometimes as \(E \cap F\), describes scenarios where both events occur at the same time. This is a crucial concept when analyzing probabilistic events that are not mutually exclusive.

Returning to a familiar die example, if E is rolling a number less than 4, and F is rolling an even number, \(EF\) would only include rolling a 2. This is the sole outcome where both conditions are fulfilled simultaneously. 

The probability of \(EF\) is only considered once when calculating the probability of the union of E and F with \(P(E \cup F) = P(E) + P(F) - P(EF)\). If not careful, failing to adjust for the intersection can lead to overestimating the combined probabilities of events.
Properties of Probabilities
Understanding the foundational properties of probabilities unlocks a deeper grasp of how probability functions work. These properties include, but are not limited to:

  • Probabilities are always between 0 and 1, that is, \(0 \leq P(E) \leq 1\). This reflects that an impossible event has no chance of occurring (0) and a certain event will always occur (1).
  • The probability of the complement of an event is calculated by subtracting the event's probability from 1, or \(P(E') = 1 - P(E)\). This is helpful when it is easier to calculate the chance of something not happening.
  • For any two events, the probability of their union can be calculated by \(P(A \cup B) = P(A) + P(B) - P(AB)\). This formula accounts for situations where events may overlap.

Understanding these basics helps in various probability scenarios, ensuring that calculations are accurate and reflect real-world possibilities.

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Most popular questions from this chapter

If two fair dice are tossed, what is the probability that the sum is \(i, i=\) \(2,3, \ldots, 12 ?\)

Um 1 contains two white balls and one black ball, while urn 2 contains one white ball and five black balls. One ball is drawn at random from urn 1 and placed in urn 2. A ball is then drawn from urn 2. It happens to be white. What is the probability that the transferred ball was white?

Bill and George go target shooting together. Both shoot at a target at the same time. Suppose Bill hits the target with probability \(0.7\), whereas George, independently, hits the target with probability \(0.4 .\) (a) Given that exactly one shot hit the target, what is the probability that it was George's shot? (b) Given that the target is hit, what is the probability that George hit it?

A coin is to be tossed until a head appears twice in a row. What is the sample space for this experiment? If the coin is fair, what is the probability that it will be tossed exactly four times?

A deck of 52 playing cards, containing all 4 aces, is randomly divided into 4 piles of 13 cards each. Define events \(E_{1}, E_{2}, E_{3}\), and \(E_{4}\) as follows: $$ \begin{aligned} &E_{1}=\\{\text { the first pile has exactly } 1 \text { ace }\\} \\ &E_{2}=\\{\text { the second pile has exactly } 1 \mathrm{ace}\\} \\ &E_{3}=\\{\text { the third pile has exactly } 1 \text { ace }\\} \end{aligned} $$ \(E_{4}=\\{\) the fourth pile has exactly 1 ace\\}
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