microfracture knee surgery has a 75 chance of success on patients with degernerative knees. the surgery is performed on 5 patients. find the probability of the surgery being succesful on less than 3 patients?
To find the probability of the surgery being successful on less than 3 patients, we need to calculate the probabilities of it being successful on 0, 1, or 2 patients, and then sum them up.
Given that the surgery has a 75% chance of success, the probability of each patient having a successful surgery is 0.75. We can now use this information to calculate the probabilities.
The probability of 0 patients having a successful surgery can be calculated using the binomial probability formula:
P(X = k) = (nCk) * p^k * (1-p)^(n-k)
Where:
- P(X = k): Probability of k successes
- n: Number of trials (number of patients in this case)
- k: Number of successful outcomes (0 in this case)
- p: Probability of success (0.75 in this case)
Using the formula, we can calculate the probability of 0 patients having a successful surgery:
P(X = 0) = (5C0) * (0.75^0) * (1 - 0.75)^(5-0)
= (1) * (1) * (0.25^5)
= 0.25^5
= 0.0009765625
Similarly, we can calculate the probabilities for 1 and 2 patients having a successful surgery using the same formula:
P(X = 1) = (5C1) * (0.75^1) * (1 - 0.75)^(5-1)
= (5) * (0.75) * (0.25^4)
= 0.0146484375
P(X = 2) = (5C2) * (0.75^2) * (1 - 0.75)^(5-2)
= (10) * (0.75^2) * (0.25^3)
= 0.087890625
Now, to find the probability of the surgery being successful on less than 3 patients, we sum up these three probabilities:
P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)
= 0.0009765625 + 0.0146484375 + 0.087890625
= 0.103515625
Therefore, the probability of the surgery being successful on less than 3 patients is approximately 0.1035, or 10.35%.