Question 1186891: In an ablation procedure, the probability of acute success (determined at completion of
the procedure) is 0.95 when an image mapping system is used. Without the image
mapping system, the probably of acute success is only0.80. Suppose that Patient A is
given the treatment with the mapping system and Patient B is given the treatment
without the mapping system. Determine the following probabilities:
a. Both patients A and B had acute successes
b. A had an acute success but B had an acute failure.
c. B had an acute success but A had an acute failure.
d. Both A and B had acute failures.
e. At least one of the patients had an acute success.
f. Describe two ways that the result in (e) can be calculated based on the results from
(a), (b), (c), and (d).
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.
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