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We are given the following probabilities:

- \( P(A) = 0.5 \)
- \( P(B) = 0.3 \)
- \( P(A \text{ OR } B) = 0.67 \)

We need to find \( P(A \text{ AND } B) \).

The formula that relates these probabilities is:

\[
P(A \text{ OR } B) = P(A) + P(B) - P(A \text{ AND } B)
\]

Substituting the given values into this equation:

\[
0.67 = 0.5 + 0.3 - P(A \text{ AND } B)
\]

Now, solve for \( P(A \text{ AND } B) \):

\[
0.67 = 0.8 - P(A \text{ AND } B)
\]

\[
P(A \text{ AND } B) = 0.8 - 0.67
\]

\[
P(A \text{ AND } B) = 0.13
\]

Thus, \( P(A \text{ AND } B) = 0.13 \).

Would you like further details or have any questions?

Here are some related questions:
1. How is the formula \( P(A \text{ OR } B) = P(A) + P(B) - P(A \text{ AND } B) \) derived?
2. What is the significance of mutually exclusive events in this formula?
3. How would the problem change if \( A \) and \( B \) were independent events?
4. Can \( P(A \text{ AND } B) \) ever be negative?
5. What is the difference between \( P(A \text{ OR } B) \) and \( P(A \cup B) \)?

**Tip:** When calculating combined probabilities, always check if the events are independent or mutually exclusive, as that can simplify the problem.

If A and B are events with P(A) = 0.5, P(B) = 0.3, P(A OR B) = 0.67, find P(A AND B).

Learn how to calculate the probability of the intersection of two events A and B given P(A), P(B), and P(A OR B) using the inclusion-exclusion principle. This problem is suitable for high school students, covering essential probability concepts.

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