Suppose A and B are events in the same sample space for some experiment. Suppose A occurs 37% of the times, B happens 5 out of 10 times the experiment is run, and there is a probability 0.41 that B occurs and A does not occur. Determine 𝑃𝑟(𝐴𝑐∪𝐵) as a decimal number with 2 decimal places.

To determine \( \Pr(A^c \cup B) \), where \( A^c \) represents the complement of \( A \), we can use the formula for the probability of the union of two events:

\[
\Pr(A^c \cup B) = \Pr(A^c) + \Pr(B) - \Pr(A^c \cap B)
\]

### Step-by-Step Solution

1. **Given Probabilities and Information**:
   - \( \Pr(A) = 0.37 \) (since \( A \) occurs 37% of the time).
   - \( \Pr(B) = 0.5 \) (since \( B \) happens 5 out of 10 times).
   - \( \Pr(B \text{ and } A^c) = 0.41 \).

2. **Calculate \( \Pr(A^c) \)**:
   Since \( A^c \) is the complement of \( A \),
   \[
   \Pr(A^c) = 1 - \Pr(A) = 1 - 0.37 = 0.63
   \]

3. **Calculate \( \Pr(A^c \cap B) \)**:
   We already know from the problem statement that \( \Pr(B \text{ and } A^c) = 0.41 \), so
   \[
   \Pr(A^c \cap B) = 0.41
   \]

4. **Calculate \( \Pr(A^c \cup B) \)**:
   Substitute the values into the formula:
   \[
   \Pr(A^c \cup B) = \Pr(A^c) + \Pr(B) - \Pr(A^c \cap B)
   \]
   \[
   = 0.63 + 0.5 - 0.41
   \]
   \[
   = 1.13 - 0.41 = 0.72
   \]

### Final Answer
\[
\Pr(A^c \cup B) = 0.72
\]

Let me know if you want more details or have any questions! 

Here are 5 related questions to expand on this information:

1. How would we find \( \Pr(A \cap B) \) using the given information?
2. What is the interpretation of \( \Pr(A^c \cap B) \) in terms of real-world scenarios?
3. How does the complement of an event help in calculating union probabilities?
4. How would the answer change if \( \Pr(A \cap B) \) were given instead of \( \Pr(A^c \cap B) \)?
5. Can we verify if \( A \) and \( B \) are independent events with this information?

**Tip:** When calculating probabilities of complements, remember that \( \Pr(A^c) = 1 - \Pr(A) \). This is often useful in problems involving unions and intersections.

Suppose A and B are events in the same sample space for some experiment. Suppose A occurs 37% of the times, B happens 5 out of 10 times the experiment is run, and there is a probability 0.41 that B occurs and A does not occur. Determine Pr(A^c ∪ B) as a decimal number with 2 decimal places.

Learn to find the probability of Pr(A^c ∪ B) using given probabilities of events A and B, including the complement rule and union of events formula. This problem covers key probability concepts, ideal for high school students studying advanced probability.

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