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 less than 3 patients
Here is the easiest way to do this problem.
Use a binomial probability table.
n = 5
x = 0, 1, 2
p = .75
Add P(0), P(1), and P(2). This will be your probability.
I hope this helps.
To find the probability of the surgery being successful on less than 3 patients out of the 5 patients, we need to consider two scenarios: when the surgery is successful on 0, 1, or 2 patients.
We can calculate the probability of each scenario and then sum them up to get the probability of the surgery being successful on less than 3 patients.
Let's calculate this step by step:
1. Probability of the surgery being successful on 0 patients:
The probability of success for each patient is 75%, meaning the probability of failure is 25%.
So, the probability of the surgery not being successful on a single patient is 25% or 0.25.
Since there are 5 patients, we want to find the probability of failure for all 5 patients, which we can calculate using the multiplication principle:
Probability of 0 successes = (0.25) * (0.25) * (0.25) * (0.25) * (0.25) = 0.25^5 = 0.001953125.
2. Probability of the surgery being successful on 1 patient:
We need to consider one successful case and four unsuccessful cases. The successful case can happen in any order, so we need to consider all the possible arrangements.
The probability of success for one patient is 75% or 0.75. And the probability of failure for one patient is 25%.
To calculate the probability of success on one patient, we multiply the probability of success by the probability of failure for the remaining four patients:
Probability of 1 success = 5C1 * (0.75 * 0.25^4) = 5 * 0.75 * 0.25^4 = 0.0146484375.
(Here, 5C1 represents "5 choose 1" which calculates the number of possible combinations of selecting 1 success from the 5 patients.)
3. Probability of the surgery being successful on 2 patients:
Again, we need to consider two successful cases and three unsuccessful cases. We need to consider all the possible arrangements.
The probability of success for two patients is (0.75 * 0.75), and the probability of failure for three patients is (0.25^3).
Probability of 2 successes = 5C2 * (0.75^2 * 0.25^3) = 10 * 0.75^2 * 0.25^3 = 0.087890625.
Finally, we sum up the probabilities from the three scenarios:
Probability of the surgery being successful on less than 3 patients = Probability of 0 successes + Probability of 1 success + Probability of 2 successes
= 0.001953125 + 0.0146484375 + 0.087890625
= 0.1044921875.
Therefore, the probability of the surgery being successful on less than 3 patients out of the 5 patients is approximately 0.104 or 10.4%.