Question

c. what is the probability that at least one of the surgeries is successful?

Explanation:

To solve this, we assume we know the probability of a single surgery failing (let's say \( P(\text{fail}) = p \)) and that the surgeries are independent.

Step 1: Define the complement event

The event "at least one successful" is the complement of "all surgeries fail". Let \( n \) be the number of surgeries. If each has a failure probability \( p \), the probability all fail is \( P(\text{all fail}) = p^n \) (by independence).

Step 2: Use the complement rule

The complement rule states \( P(\text{at least one success}) = 1 - P(\text{all fail}) \).

For example, if there are 2 surgeries with \( P(\text{success}) = 0.8 \) (so \( P(\text{fail}) = 0.2 \)):

• \( P(\text{all fail}) = 0.2 \times 0.2 = 0.04 \)
• \( P(\text{at least one success}) = 1 - 0.04 = 0.96 \)

To provide a numerical answer, we need the number of surgeries and the probability of a single surgery failing (or succeeding). If we assume 2 surgeries with \( P(\text{success}) = 0.8 \) (as above), the answer is \( 0.96 \).

(Note: Replace with actual values from the problem’s context if provided.)

Answer:

\( 1 - P(\text{all surgeries fail}) \) (or a numerical value like \( 0.96 \) with specific inputs)