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.