Select the most appropriate response.

An event A will occur with probability 0.4. An event B will occur with probability 0.6. The probability that both A and B will occur is 0.20. We may conclude

Question 9 options:

that events A and B are independent.


that events A and B are dependent.


that either A or B always occurs.


that events A and B are mutually exclusive.



To determine the correct conclusion, we can check whether events A and B are independent. For two events to be independent, the probability that both occur must equal the product of their individual probabilities, i.e.,

\[
P(A \cap B) = P(A) \times P(B)
\]

Let's verify this using the given information:

- \( P(A) = 0.4 \)
- \( P(B) = 0.6 \)
- \( P(A \cap B) = 0.2 \)

Calculate the product of \( P(A) \) and \( P(B) \):

\[
P(A) \times P(B) = 0.4 \times 0.6 = 0.24
\]

Since \( P(A \cap B) = 0.2 \), which is not equal to \( 0.24 \), events A and B are not independent. This means we can rule out the first option.

Next, events are mutually exclusive if they cannot occur together, which means \( P(A \cap B) \) would be zero. Since \( P(A \cap B) = 0.2 \), events A and B are not mutually exclusive, so we can rule out the last option as well.

There is no indication that either A or B always occurs, as both have probabilities less than 1, so the third option can also be ruled out.

Thus, the most appropriate response is:

**Events A and B are dependent.**

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Would you like further clarification or details? Here are some related questions for deeper understanding:

1. What is the formula for the probability of the union of two events?
2. How does the concept of conditional probability relate to dependent events?
3. Can you have events that are both independent and mutually exclusive?
4. How can we calculate the probability of either event A or B occurring?
5. What real-world examples can represent dependent events?

**Tip**: When dealing with probabilities, always check whether the given values satisfy the conditions for independence.

An event A will occur with probability 0.4. An event B will occur with probability 0.6. The probability that both A and B will occur is 0.20. We may conclude: 

- that events A and B are independent.
- that events A and B are dependent.
- that either A or B always occurs.
- that events A and B are mutually exclusive.

Explore the probability problem involving events A and B, with given probabilities of occurrence. Learn how to analyze whether events are independent, dependent, or mutually exclusive using the formula P(A ∩ B) = P(A) × P(B). Suitable for Grades 10-12.

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