Microfracture knee surgery has a 75% chance of success on patients with degenerative knees. The surgery is performed on 5 patients. Find the probability of the surgery being successful on 3 or 4 patients?

To find the probability of the surgery being successful on 3 or 4 patients out of the 5, we need to calculate the probability for each combination separately and then add them up.

The probability of the surgery being successful on 3 patients can be calculated using the binomial probability formula:

P(X=k) = (n choose k) * p^k * q^(n-k)

Where:
- k is the number of successes (3 in this case)
- n is the total number of trials (5 in this case)
- p is the probability of success on each trial (0.75 in this case)
- q is the probability of failure on each trial (1 - p = 0.25 in this case)

Using the formula, the probability of the surgery being successful on 3 patients is:

P(X=3) = (5 choose 3) * (0.75^3) * (0.25^(5-3))

P(X=3) = (5!/(3!(5-3)!)) * (0.75^3) * (0.25^2)

P(X=3) = (10) * (0.421875) * (0.0625)

P(X=3) ≈ 0.2659

Similarly, we can calculate the probability of success on 4 patients:

P(X=4) = (5 choose 4) * (0.75^4) * (0.25^(5-4))

P(X=4) = (5!/(4!(5-4)!)) * (0.75^4) * (0.25^1)

P(X=4) = (5) * (0.421875) * (0.25)

P(X=4) ≈ 0.2637

To find the probability of the surgery being successful on 3 or 4 patients, we add the probabilities together:

P(X=3 or X=4) ≈ 0.2659 + 0.2637

P(X=3 or X=4) ≈ 0.5296

Therefore, the probability of the surgery being successful on 3 or 4 patients out of the 5 is approximately 0.5296 or 52.96%.