If ​P(A)equals0.8​, ​P(B)equals0.9​, and A and B are​ independent, find​ P(A and​ B).
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Part 1
​P(A and ​B)equals

Since events \( A \) and \( B \) are independent, the probability of both \( A \) and \( B \) occurring together, denoted \( P(A \text{ and } B) \), can be found using the formula for the intersection of two independent events:

\[
P(A \text{ and } B) = P(A) \times P(B)
\]

Given:
- \( P(A) = 0.8 \)
- \( P(B) = 0.9 \)

Now, calculate:

\[
P(A \text{ and } B) = 0.8 \times 0.9 = 0.72
\]

Thus, \( P(A \text{ and } B) = 0.72 \).

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the formula for the union of two independent events?
2. How does dependence between events affect the calculation of \( P(A \text{ and } B) \)?
3. If \( P(A) \) and \( P(B) \) were not independent, how would you calculate \( P(A \text{ and } B) \)?
4. What does it mean for two events to be mutually exclusive?
5. How do you calculate \( P(A \text{ or } B) \) for independent events?

**Tip:** Independent events do not influence each other's outcomes, so their joint probability is always the product of their individual probabilities.

If P(A) = 0.8, P(B) = 0.9, and A and B are independent, find P(A and B).

Learn how to calculate the probability of two independent events with P(A) = 0.8 and P(B) = 0.9 using the multiplication rule for independent events. This problem demonstrates key concepts in probability and is suitable for students in Grades 9-12.

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