Q12 If A and B are two independent events with p(A) = p(B) = k. What is the probability of

occurrence of at least one of these two events?

a)k2 b) 2k 􀀀 k2 c) k2 􀀀 2k d) 2 􀀀 k

To find the probability of the occurrence of at least one of the two independent events \( A \) and \( B \), we use the following formula:

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

Since \( A \) and \( B \) are independent events, the probability of both events occurring together (\( P(A \cap B) \)) is given by:

\[
P(A \cap B) = P(A) \times P(B) = k \times k = k^2
\]

Thus, the probability of at least one of these two events occurring is:

\[
P(A \cup B) = k + k - k^2 = 2k - k^2
\]

So, the correct answer is **b) \( 2k - k^2 \)**.

Would you like more details or have any questions about this solution? Here are some related questions you might find interesting:

1. What is the formula for the probability of the union of two independent events?
2. How do you calculate the probability of two independent events occurring together?
3. What is the difference between independent and mutually exclusive events?
4. How does the probability of the union of events change if the events are not independent?
5. What does it mean for two events to be independent?
6. How do you calculate the probability of at least one event occurring?
7. What happens to the probability of the union of events as \( k \) approaches 1?
8. Can the probability of at least one event be greater than 1?

**Tip:** When dealing with independent events, always remember that the occurrence of one event does not affect the probability of the other event.

Understand how to calculate the probability of independent events A and B occurring together or individually. This problem covers the formula for the probability of the union of events and its application in determining the probability of at least one event. Suitable for students studying probability theory in Grades 10-12.

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