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?

---

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.