Suppose P(A)=0.58 , P(B|A) =0.71 , P(C|A and B)=0.79 . Determine P( A and B and C ).

To determine \( P(A \cap B \cap C) \), we can use the definition of conditional probability in a step-by-step process.

We know the following:
- \( P(A) = 0.58 \)
- \( P(B | A) = 0.71 \)
- \( P(C | A \cap B) = 0.79 \)

The joint probability of \( A \cap B \cap C \) is calculated using the product rule of probability:

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

Substitute the given values:

\[
P(A \cap B \cap C) = 0.58 \times 0.71 \times 0.79
\]

Let me compute this for you.The probability of \( A \cap B \cap C \) (i.e., \( P(A \cap B \cap C) \)) is approximately \( 0.325 \).

Would you like further details or have any other questions?

Here are 5 related questions to consider:
1. How would the result change if \( P(C|A \cap B) \) were different?
2. What is the interpretation of conditional probability in real-world scenarios?
3. How can this concept be extended to more than three events?
4. What is the probability of \( P(A \cap B) \)?
5. How does Bayes' Theorem relate to conditional probability?

**Tip:** Always ensure conditional probabilities add up consistently with the base probabilities when solving problems.

Suppose P(A)=0.58, P(B|A)=0.71, P(C|A and B)=0.79. Determine P(A and B and C).

Learn how to calculate the joint probability P(A ∩ B ∩ C) using conditional probabilities with the given values: P(A) = 0.58, P(B|A) = 0.71, and P(C|A ∩ B) = 0.79. This step-by-step solution uses the product rule of probability. Ideal for students in Grades 10-12.

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