Probability of an Event and its Complement

Oct 13, 2025 by miodeya.com 27 views

In probability theory, the concept of complement is a fundamental idea that helps us understand the relationship between two events. The complement of an event is the set of all outcomes that are not part of the original event. In this article, we will explore the concept of complement in probability and how it can be used to find the probability of an event's complement.

What is the Complement of an Event?

The complement of an event is the set of all outcomes that are not part of the original event. For example, if we flip a coin, the event "heads" has a complement "tails." If we roll a die, the event "even number" has a complement "odd number."

Probability of an Event and its Complement

The probability of an event and its complement are related in a way that their sum is equal to 1. This is known as the complement rule in probability theory. Mathematically, it can be expressed as:

P(A) + P(A') = 1

where P(A) is the probability of event A, and P(A') is the probability of the complement of event A.

Finding the Probability of an Event's Complement

Now, let's apply the complement rule to find the probability of an event's complement. We are given that the probability of an event is $\frac{2}{7}$ . We need to find the probability of its complement.

Using the complement rule, we can write:

P(A) + P(A') = 1

Substituting the given value, we get:

$\frac{2}{7}$ + P(A') = 1

To find the probability of the complement, we need to isolate P(A').

P(A') = 1 - $\frac{2}{7}$

P(A') = $\frac{7}{7}$ - $\frac{2}{7}$

P(A') = $\frac{5}{7}$

Therefore, the probability of the event's complement is $\frac{5}{7}$ .

Conclusion

In this article, we explored the concept of complement in probability and how it can be used to find the probability of an event's complement. We applied the complement rule to find the probability of an event's complement, given that the probability of the event is $\frac{2}{7}$ . The result shows that the probability of the event's complement is $\frac{5}{7}$ .

Answer

The correct answer is D. $\frac{5}{7}$ .

Key Takeaways

The complement of an event is the set of all outcomes that are not part of the original event.
The probability of an event and its complement are related in a way that their sum is equal to 1.
The complement rule can be used to find the probability of an event's complement.
The probability of an event's complement is 1 minus the probability of the event.
Frequently Asked Questions (FAQs) about Complement in Probability ====================================================================

In this article, we will address some of the most common questions related to the concept of complement in probability.

Q: What is the complement of an event?

A: The complement of an event is the set of all outcomes that are not part of the original event. For example, if we flip a coin, the event "heads" has a complement "tails." If we roll a die, the event "even number" has a complement "odd number."

Q: How is the probability of an event related to its complement?

A: The probability of an event and its complement are related in a way that their sum is equal to 1. This is known as the complement rule in probability theory. Mathematically, it can be expressed as:

P(A) + P(A') = 1

where P(A) is the probability of event A, and P(A') is the probability of the complement of event A.

Q: How do I find the probability of an event's complement?

A: To find the probability of an event's complement, you can use the complement rule. If you know the probability of the event, you can subtract it from 1 to find the probability of the complement.

P(A') = 1 - P(A)

Q: What is the difference between the complement and the union of two events?

A: The complement of an event is the set of all outcomes that are not part of the original event. The union of two events, on the other hand, is the set of all outcomes that are part of either event. For example, if we have two events A and B, the union of A and B is the set of all outcomes that are part of A or B or both.

Q: Can the complement of an event be the same as the event itself?

A: No, the complement of an event cannot be the same as the event itself. By definition, the complement of an event is the set of all outcomes that are not part of the original event. If the complement of an event is the same as the event itself, then it would mean that the event has no outcomes, which is not possible.

Q: What is the relationship between the probability of an event and its complement in the case of mutually exclusive events?

A: In the case of mutually exclusive events, the probability of the complement of one event is equal to the probability of the other event. This is because the two events are mutually exclusive, meaning that they cannot occur at the same time.

Q: Can the probability of an event's complement be greater than 1?

A: No, the probability of an event's complement cannot be greater than 1. By definition, the probability of an event and its complement must add up to 1.

Q: What is the significance of the complement rule in probability theory?

A: The complement rule is a fundamental concept in probability theory that helps us understand the relationship between an event and its complement. It is used to find the probability of an event's complement and is a crucial tool in probability calculations.

Conclusion

In this article, we addressed some of the most common questions related to the concept of complement in probability. We hope that this FAQ section has provided you with a better understanding of the concept and its applications in probability theory.