Let A and B be any two events with probability P(A)=P(B)=0.9 and AB
denotes the intersection of A and B. If P(A'UB')=0.2, then P(AB' U A'B)=

We are given the following information:

- \( P(A) = 0.9 \)
- \( P(B) = 0.9 \)
- \( P(A' \cup B') = 0.2 \)

We are tasked to find \( P(AB' \cup A'B) \), which represents the probability that either \( A \) occurs but \( B \) does not, or \( B \) occurs but \( A \) does not. This is equivalent to \( P(A \Delta B) \) (the symmetric difference of \( A \) and \( B \)).

### Step 1: Relation between the complements and union
We know that:
\[
P(A' \cup B') = 1 - P(A \cap B)
\]
Thus:
\[
P(A \cap B) = 1 - P(A' \cup B') = 1 - 0.2 = 0.8
\]

### Step 2: Decompose the symmetric difference
The event \( AB' \cup A'B \) corresponds to the symmetric difference between \( A \) and \( B \). The formula for the symmetric difference is:
\[
P(A \Delta B) = P(A) + P(B) - 2P(A \cap B)
\]
Substitute the known values:
\[
P(A \Delta B) = 0.9 + 0.9 - 2 \times 0.8 = 1.8 - 1.6 = 0.2
\]

Thus, the probability \( P(AB' \cup A'B) = 0.2 \).

### Final Answer:
\[
P(AB' \cup A'B) = 0.2
\]

Do you want more details or have any questions?

Here are 5 related questions to explore further:
1. How do you calculate the union of two independent events?
2. What is the meaning of the symmetric difference in probability theory?
3. How does \( P(A \cup B) \) relate to the individual probabilities of \( A \) and \( B \)?
4. What happens if \( A \) and \( B \) are mutually exclusive events?
5. How does Bayes' theorem relate to conditional probabilities of intersections?

**Tip:** When solving problems involving multiple events, Venn diagrams can be a helpful visual tool to understand the relationships between intersections and unions.

Let A and B be any two events with probability P(A)=P(B)=0.9 and AB denotes the intersection of A and B. If P(A' U B')=0.2, then P(AB' U A'B) = ?

This problem involves finding P(AB' U A'B) using set theory and probability concepts. Given that P(A) = 0.9, P(B) = 0.9, and P(A' U B') = 0.2, this problem applies the symmetric difference theorem. Ideal for students in Grades 11-12 or college-level probability courses.

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