Problem 7 - A Modern Introduction to Probability and Statistics: Understanding Why and How

Question

Let \(A\) and \(B\) be two events. Suppose that \(\mathrm{P}(A)=0.4, \mathrm{P}(B)=0.5\), and \(\mathrm{P}(A \cap B)=0.1\). Find the probability that \(A\) or \(B\) occurs, but not both.

Step-by-Step Solution

Answer

The probability is 0.7.

Step 1: Understanding the Problem

We are given the probabilities of two events: \(A\) and \(B\), and their intersection. Our goal is to find the probability of the exclusive event \(A\) or \(B\), which means one occurs but not both.

Step 2: Applying the Formula for Union of Two Events

We use the formula for the union of two events \(A\) and \(B\): \[\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B)\]Substitute the given values:\[\mathrm{P}(A \cup B) = 0.4 + 0.5 - 0.1 = 0.8.\]

Step 3: Finding the Probability of Both Events Happening

Since we want either \(A\) or \(B\), but not both, we first determine the probability of \(A\) and \(B\) both occurring which is given as \(\mathrm{P}(A \cap B) = 0.1.\)

Step 4: Subtracting the Intersection from the Union

The probability that either \(A\) or \(B\) occurs, but not both, can be found by subtracting the intersection from the union probability derived in the second step:\[\mathrm{P}((A \cup B) \cap eg(A \cap B)) = \mathrm{P}(A \cup B) - \mathrm{P}(A \cap B), \]which gives us\[0.8 - 0.1 = 0.7.\]

Step 5: Conclusion of Solution

Thus, the probability that either event \(A\) or event \(B\) occurs, but not both, is \(0.7\).

Key Concepts

Events

In probability theory, an "event" is a fundamental concept that represents a particular outcome or a set of outcomes from a random experiment. Each event is a subset of all possible outcomes, known as the sample space. For example, if we toss a single die, an event could be rolling an even number. Some key points about events include:

• Events can be as specific as a single outcome (like rolling a three) or more general (like rolling an even number).
• Events are often denoted by capital letters such as \(A\), \(B\), etc.
• The probability of an event represents the likelihood that the event will occur during the experiment.

Probability values range from 0 (impossible event) to 1 (certain event). Understanding events helps us to calculate and analyze the probabilities associated with different outcomes.

Intersection

When we talk about intersection in probability, we refer to the event in which two or more events occur simultaneously. The intersection of events \(A\) and \(B\) is denoted by \(A \cap B\). This set includes only the outcomes that events \(A\) and \(B\) have in common.
It's essential to remember:

• The probability of the intersection is symbolized as \(\mathrm{P}(A \cap B)\).
• If the events are independent, the probability of their intersection is the product of their individual probabilities: \(\mathrm{P}(A \cap B) = \mathrm{P}(A) \cdot \mathrm{P}(B)\).
• In cases of dependence, this simple multiplication may not apply. Additional data about the dependency is required.

The intersection gives insight into the shared occurrence of events, which is crucial for calculating exclusive probabilities.

Union

The union of events \(A\) and \(B\) refers to the event that occurs if either \(A\) occurs, \(B\) occurs, or both occur. The union of two events is represented as \(A \cup B\).
This concept is essential for assessing the combined likelihood of two events. The probability of the union is given by the formula:
\[\mathrm{P}(A \cup B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cap B) \]
Key points about the union:

• It's crucial to subtract \(\mathrm{P}(A \cap B)\) to avoid double-counting the overlap where both events occur.
• This formula helps calculate the probability of at least one of the events happening.
• Union is relevant when evaluating non-mutually exclusive events, ensuring that we account for multiple possibilities.

Understanding the union is vital for comprehensive probability analysis.

Exclusive Probability

Exclusive probability is a specific probability calculation where we determine the likelihood of one event occurring without the occurrence of another. In terms of sets, this uses the concept of the symmetric difference. To find the probability that either \(A\) or \(B\) occurs but not both, we use:
\[\mathrm{P}((A \cup B) \cap eg(A \cap B)) = \mathrm{P}(A \cup B) - \mathrm{P}(A \cap B)\]
What this tells us is:

• This formula accounts for the combined events and removes cases where both occur.
• It's also known as calculating the probability of an "exclusive or," where at least one event occurs, but not the intersection.
• An understanding of both union and intersection helps find this value.

This exclusive probability measure is beneficial for scenarios where dual occurrences are not desired.