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Assume P(A) = .3 and P(B) = .4. If the P(A or B) = .7 we can say that A and B are:

Multiple select question.

disjoint.

compliments.

mutually exclusive.

independent.

Given the probabilities \( P(A) = 0.3 \), \( P(B) = 0.4 \), and \( P(A \text{ or } B) = 0.7 \), we can determine the relationship between events A and B.

1. **Disjoint (Mutually Exclusive)**: Events A and B are disjoint or mutually exclusive if they cannot occur at the same time, i.e., \( P(A \cap B) = 0 \). For disjoint events, \( P(A \text{ or } B) = P(A) + P(B) \). Here, \( P(A) + P(B) = 0.3 + 0.4 = 0.7 \), which matches \( P(A \text{ or } B) = 0.7 \). Therefore, A and B are disjoint or mutually exclusive.

2. **Complements**: Events A and B are complements if \( P(A) + P(B) = 1 \) and \( P(A \cap B) = 0 \). Since \( P(A) + P(B) = 0.7 \neq 1 \), A and B are not complements.

3. **Independent**: Events A and B are independent if \( P(A \cap B) = P(A) \times P(B) \). Here, \( P(A) \times P(B) = 0.3 \times 0.4 = 0.12 \). We know \( P(A \text{ or } B) = P(A) + P(B) - P(A \cap B) = 0.7 \). This implies \( P(A \cap B) = 0.7 - 0.7 = 0 \). Since \( 0 \neq 0.12 \), A and B are not independent.

Based on the given probabilities, we can conclude that A and B are:

- Disjoint (Mutually Exclusive)

Would you like more details or have any questions?

Here are some questions you might ask next:
1. How do you determine if events are disjoint?
2. What is the difference between disjoint and independent events?
3. How can you find the probability of the intersection of two events?
4. What does it mean for events to be complements?
5. How does the formula for \( P(A \text{ or } B) \) change if events are not disjoint?

**Tip:** When dealing with probabilities of combined events, always check if the events are disjoint or independent, as this affects how you calculate their combined probabilities.

Explore the concept of disjoint events A and B in probability. Given probabilities P(A) = 0.3, P(B) = 0.4, and P(A or B) = 0.7, learn how to determine if events are disjoint or mutually exclusive. This problem involves key concepts in probability theory and set theory, suitable for advanced high school students.

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