The Formula for Union of Two Events

Oct 13, 2025 by windailys.com 108 views

Introduction

Probability is a fundamental concept in mathematics that deals with the study of chance events. It is used to measure the likelihood of occurrence of an event. In this article, we will discuss the probability of union of two events, which is a crucial concept in probability theory. We will use the given information to find the indicated probability.

Understanding the Problem

The problem provides us with the following information:

$P(C) = 0.048$
$P(M \cap C) = 0.044$
$P(M \cup C) = 0.524$

We are asked to find the indicated probability $P[(M \cup C)]$ .

The Formula for Union of Two Events

The formula for the union of two events is given by:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

where $P(A \cup B)$ is the probability of the union of events $A$ and $B$ , $P(A)$ is the probability of event $A$ , $P(B)$ is the probability of event $B$ , and $P(A \cap B)$ is the probability of the intersection of events $A$ and $B$ .

Applying the Formula

We can apply the formula for the union of two events to the given information:

$P(M \cup C) = P(M) + P(C) - P(M \cap C)$

We are given that $P(M \cup C) = 0.524$ , $P(C) = 0.048$ , and $P(M \cap C) = 0.044$ . We need to find $P(M)$ .

Finding the Probability of Event M

We can rearrange the formula to solve for $P(M)$ :

$P(M) = P(M \cup C) - P(C) + P(M \cap C)$

Substituting the given values, we get:

$P(M) = 0.524 - 0.048 + 0.044$

$P(M) = 0.520$

Finding the Indicated Probability

Now that we have found $P(M)$ , we can find the indicated probability $P[(M \cup C)]$ .

Since $P(M \cup C) = 0.524$ , we can conclude that the indicated probability is also $0.524$ .

Conclusion

In this article, we discussed the probability of union of two events. We used the given information to find the indicated probability. We applied the formula for the union of two events and solved for the probability of event $M$ . Finally, we found the indicated probability, which is $0.524$ .

Answer

The correct answer is:

D. 0.476

Q&A: Probability of Union of Two Events

Q: What is the probability of union of two events?

A: The probability of union of two events is the probability that at least one of the events will occur. It is denoted by $P(A \cup B)$ .

Q: How is the probability of union of two events calculated?

A: The probability of union of two events is calculated using the formula:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Q: What is the difference between the probability of union and the probability of intersection?

A: The probability of union is the probability that at least one of the events will occur, while the probability of intersection is the probability that both events will occur.

Q: Can you give an example of how to calculate the probability of union of two events?

A: Suppose we have two events $A$ and $B$ , and we know that:

$P(A) = 0.3$
$P(B) = 0.4$
$P(A \cap B) = 0.1$

We can calculate the probability of union of two events using the formula:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

$P(A \cup B) = 0.3 + 0.4 - 0.1$

$P(A \cup B) = 0.6$

Q: What is the relationship between the probability of union and the probability of intersection?

A: The probability of union is always greater than or equal to the probability of intersection. This is because the union of two events includes all the outcomes that are in either event, while the intersection of two events includes only the outcomes that are in both events.

Q: Can you give an example of how to use the formula for the probability of union of two events?

A: Suppose we have two events $M$ and $C$ , and we know that:

$P(M \cup C) = 0.524$
$P(C) = 0.048$
$P(M \cap C) = 0.044$

We can use the formula to find the probability of event $M$ :

$P(M) = P(M \cup C) - P(C) + P(M \cap C)$

$P(M) = 0.524 - 0.048 + 0.044$

$P(M) = 0.520$

Q: What is the indicated probability in the given problem?

A: The indicated probability is $P[(M \cup C)]$ , which is equal to $0.524$ .

Conclusion

In this article, we discussed the probability of union of two events. We answered some common questions related to the probability of union of two events, including how to calculate it, the difference between the probability of union and the probability of intersection, and how to use the formula for the probability of union of two events. We also gave an example of how to use the formula to find the probability of event $M$ .