Question

Find Binomial Probability by using Formula P(x)=,C,p* q∗−ᵏ= n!/(n−x)∣x p²q*..

Rotator cuff surgery has a 80% chance of success. The surgery is performed on four patients. Find the probability of the surgery being successful on exactly two patients. (Source: The Orthopedic Center of St. Louis)

Answer 1

To find the probability of exactly two successful rotator cuff surgeries out of four, we use the binomial probability formula with a success rate of 80%. The probability is 15.36%.

Step-by-step explanation:

The student's question is about finding the probability of a binomial experiment where the surgeries have an 80% chance of success and are performed on four patients. To calculate the probability of exactly two successful surgeries, we can use the binomial probability formula:

P(X = x) = C(n, x) * p^x * q^(n-x)

Where:

• n is the number of trials (in this case, 4)
• p is the probability of success (0.80)
• q is the probability of failure (1 - p = 0.20)
• x is the number of successes we're finding the probability for (2 in this instance)

The combination formula C(n, x) equals n! / (x!(n-x)!) where '!' denotes factorial. The calculation is as follows:

1. Calculate the combination for x successes in n trials: C(4, 2) = 4! / (2! * (4-2)!) = 6
2. Calculate p^x: (0.80)^2 = 0.64
3. Calculate q^(n-x): (0.20)^2 = 0.04
4. Multiply them together: 6 * 0.64 * 0.04 = 0.1536

Therefore, the binomial probability of exactly two successes out of four trials is 0.1536, or 15.36%.