Consider two events A and B.
 
You know that the probability of B given A is 0.29, and the probability of B given that A has not occured is 0.27. 
 
Also it is known that the probability of A occuring is 0.26. 
 
Calculate to four decimal places the probility that A has occured given that B occurs. 

To calculate the probability that event \(A\) has occurred given that event \(B\) occurs, denoted as \(P(A \mid B)\), we can use **Bayes' Theorem**. 

Bayes' Theorem is formulated as:

\[
P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}
\]

where:
- \(P(A \mid B)\) is the probability of \(A\) given \(B\),
- \(P(B \mid A)\) is the probability of \(B\) given \(A\),
- \(P(A)\) is the probability of \(A\),
- \(P(B)\) is the probability of \(B\).

### Given Information

- \(P(B \mid A) = 0.29\): the probability of \(B\) given \(A\).
- \(P(B \mid A^c) = 0.27\): the probability of \(B\) given \(A\) has not occurred (where \(A^c\) denotes the complement of \(A\)).
- \(P(A) = 0.26\): the probability of \(A\).

We need to find \(P(A \mid B)\).

### Step-by-Step Solution

1. **Find \(P(B)\) using the Law of Total Probability**:

   The Law of Total Probability states that if \(A\) and \(A^c\) are mutually exclusive and exhaustive events, then:

   \[
   P(B) = P(B \mid A) \cdot P(A) + P(B \mid A^c) \cdot P(A^c)
   \]

   Here, \(P(A^c) = 1 - P(A)\).

   So, we calculate \(P(B)\) as:

   \[
   P(B) = (0.29) \cdot (0.26) + (0.27) \cdot (1 - 0.26)
   \]

   Let's calculate \(P(B)\):

\[
P(B) = (0.29 \times 0.26) + (0.27 \times 0.74)
\]

2. **Compute \(P(B)\)**:

\[
P(B) = (0.0754) + (0.1998)
\]

\[
P(B) = 0.2752
\]

3. **Apply Bayes' Theorem to find \(P(A \mid B)\)**:

\[
P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}
\]

Substitute the values:

\[
P(A \mid B) = \frac{0.29 \cdot 0.26}{0.2752}
\]

Now calculate:

\[
P(A \mid B) = \frac{0.0754}{0.2752}
\]

\[
P(A \mid B) \approx 0.2741
\]

### Final Answer

The probability that \(A\) has occurred given that \(B\) occurs is **0.2741** to four decimal places.

Would you like more details, or do you have any questions?

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Here are 5 related questions for further exploration:

1. What is the probability that event \(A^c\) occurs given that event \(B\) occurs?
2. How would the probability \(P(A \mid B)\) change if \(P(B \mid A)\) was 0.5 instead of 0.29?
3. How can you use the complement rule to verify your calculations?
4. What would the calculation look like if events \(A\) and \(B\) were independent?
5. How does Bayes' Theorem apply to continuous probability distributions?

**Tip:** Remember that Bayes' Theorem is very powerful for updating probabilities when new information is available, particularly in real-world applications like medical testing and machine learning.

Explore how to calculate the probability that event A has occurred given that event B occurs using Bayes' Theorem. Understand the concepts of conditional probability and the law of total probability. Detailed step-by-step solution with numerical examples.

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