https://file.aiquickdraw.com/mathbot/file/1727212465673q6546iqs.jpg

The question asks for the probability that at least one of events \( A \) or \( B \) occurs, given that they are independent, with the probabilities:

- \( P(A) = 0.9 \)
- \( P(B) = 0.4 \)

To calculate the probability that at least one of \( A \) or \( B \) occurs (i.e., \( P(A \cup B) \)), we use the formula:

\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

Since \( A \) and \( B \) are independent, we can calculate \( P(A \cap B) \) as:

\[
P(A \cap B) = P(A) \times P(B) = 0.9 \times 0.4 = 0.36
\]

Now substitute into the formula for \( P(A \cup B) \):

\[
P(A \cup B) = 0.9 + 0.4 - 0.36 = 1.3 - 0.36 = 0.94
\]

Thus, the probability that at least one of the events \( A \) or \( B \) occurs is \( 0.94 \), which matches the answer in the image.

Would you like more details on how independence affects the probability calculation or have any other questions?

### Related Questions:
1. How would the result change if events \( A \) and \( B \) were not independent?
2. What is the probability that neither \( A \) nor \( B \) occurs?
3. How is the formula for \( P(A \cup B) \) derived in general?
4. What happens if the probabilities \( P(A) \) and \( P(B) \) were switched?
5. Can you explain how conditional probability works with independent events?

**Tip:** Always check whether events are independent or mutually exclusive when calculating combined probabilities.

Let A and B be independent events with P(A) = 0.9 and P(B) = 0.4. What is the probability that at least one of A or B will occur? Enter your answer in decimal form and round to four decimal places if rounding is necessary.

Learn how to calculate the probability of at least one of two independent events occurring, given P(A) = 0.9 and P(B) = 0.4. The solution involves the inclusion-exclusion principle and independence of events. Step-by-step explanation suitable for high school students in Grades 9-12.

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