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.