Question 1186891
Here's how to calculate the probabilities:

Let's define the events:

*   A: Patient A has acute success.
*   B: Patient B has acute success.

We are given:

*   P(A) = 0.95 (Probability of success with mapping system)
*   P(B) = 0.80 (Probability of success without mapping system)

Since the treatments are independent for each patient, we can multiply the probabilities.

**a. Both patients A and B had acute successes:**

P(A and B) = P(A) * P(B) = 0.95 * 0.80 = 0.76

**b. A had an acute success but B had an acute failure:**

P(A and not B) = P(A) * P(not B) = 0.95 * (1 - 0.80) = 0.95 * 0.20 = 0.19

**c. B had an acute success but A had an acute failure:**

P(not A and B) = P(not A) * P(B) = (1 - 0.95) * 0.80 = 0.05 * 0.80 = 0.04

**d. Both A and B had acute failures:**

P(not A and not B) = P(not A) * P(not B) = (1 - 0.95) * (1 - 0.80) = 0.05 * 0.20 = 0.01

**e. At least one of the patients had an acute success:**

There are two ways to calculate this:

*   **Method 1 (Direct):**  This is the complement of *both* having acute failures.

    P(at least one success) = 1 - P(not A and not B) = 1 - 0.01 = 0.99

*   **Method 2 (Addition):**  Add the probabilities of the scenarios where at least one has a success.

    P(at least one success) = P(A and B) + P(A and not B) + P(not A and B) = 0.76 + 0.19 + 0.04 = 0.99

**f. Two ways to calculate (e):**

As shown above:

1.  **Complement Rule:** The probability of at least one success is 1 minus the probability that *neither* patient has a success (calculated in part d).

2.  **Addition Rule (for mutually exclusive events):** Add the probabilities of all the scenarios where at least one patient has a success (calculated in parts a, b, and c).  These scenarios are mutually exclusive, so we can simply add their probabilities.