To find $P(A' \cup B)$, we can use the complementary rule and set theory properties.

Given:

1. $P(A) = 0.55$
2. $P(B) = 0.35$
3. $P(A \cup B) = 0.75$

Solution:

Step 1: Complement Rule

The event $A' \cup B$ (the complement of $A$ union $B$) can be expressed as:

$P(A' \cup B) = 1 - P(A \cap B').$

Step 2: Decompose $P(A \cup B)$

Using the formula for the union of two events:

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

Substitute the given values:

$0.75 = 0.55 + 0.35 - P(A \cap B).$

Solve for $P(A \cap B)$:

\