Refer to the Venn diagram to the right for events A and B in an equally likely sample space S. Find the indicated probability.

​P(Aunion∪​B)

10352431

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Part 1

Complete the Venn diagram below with each probability.

S

A

B

. 1.1

. 35.35

. 24.24

. 31.31

​(Type integers or​ decimals.)

Part 2

​P(Aunion∪​B)equals=enter your response here

​(Type an integer or a​ decimal.)

To solve this, let's first understand the elements provided and the structure of a Venn diagram for sets \( A \) and \( B \) in a sample space \( S \).

### Part 1: Completing the Venn Diagram

We are given probabilities for different regions of the Venn diagram. The regions are typically divided into four parts:
1. The part of \( A \) only (outside of \( B \)).
2. The part of \( B \) only (outside of \( A \)).
3. The intersection of \( A \) and \( B \) (i.e., both \( A \) and \( B \)).
4. The part of \( S \) outside of both \( A \) and \( B \).

We are provided the following values:
- 1.1: likely refers to the part of \( S \) outside both \( A \) and \( B \).
- 35.35: likely refers to the part of \( A \) only.
- 24.24: likely refers to the intersection \( A \cap B \).
- 31.31: likely refers to the part of \( B \) only.

### Part 2: Finding \( P(A \cup B) \)

The probability of the union \( A \cup B \) is the sum of the probabilities of the areas that include \( A \), \( B \), or both:
\[
P(A \cup B) = P(\text{A only}) + P(\text{B only}) + P(A \cap B)
\]
Using the values from the problem:
\[
P(A \cup B) = 35.35 + 31.31 + 24.24
\]
Let’s calculate the sum.

\[
P(A \cup B) = 35.35 + 31.31 + 24.24 = 90.9
\]

Thus, \( P(A \cup B) = 90.9 \).

### Summary:
1. The Venn diagram is filled with these values:
   - Region outside \( A \cup B \): 1.1
   - Region \( A \) only: 35.35
   - Region \( B \) only: 31.31
   - Region \( A \cap B \): 24.24
2. The probability of the union \( A \cup B \) is \( 90.9 \).

Would you like a deeper explanation on how these regions are identified? Also, here are 5 related questions for further practice:

1. How do you calculate the probability of \( A \cap B \) from a Venn diagram?
2. What is the probability of the complement of \( A \cup B \)?
3. How would you interpret the probability of \( A \cup B \) in terms of real-world events?
4. If we knew \( P(A) \) and \( P(B) \), how could we verify if events \( A \) and \( B \) are independent?
5. How would the Venn diagram change if \( P(A \cap B) = 0 \)?

**Tip:** The probability of the union \( A \cup B \) is always the sum of the individual probabilities minus their overlap (intersection). This prevents double-counting the intersection.

Refer to the Venn diagram to the right for events A and B in an equally likely sample space S. Find the indicated probability.

P(A ∪ B)

Complete the Venn diagram below with each probability.

S
A
B

1.1
35.35
24.24
31.31

(Type integers or decimals.)

P(A ∪ B) = (Type an integer or a decimal.)

Learn how to solve a probability problem using a Venn diagram with events A and B in an equally likely sample space. This detailed guide covers set theory, probability calculations using the inclusion-exclusion principle, and provides step-by-step instructions. Suitable for students in Grades 9-12.

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