A surgical technique is performed on 7 patients. There is a 70% chance of success. Find the probability that the surgery is successful for at least 5 patients and what is the expected number of successes out of 20 surgeries.
My attempt was: P(xmore than or equal to 5)= P(x=5)+ P(x=6) + P(x=7)
and expected number of success for 20 surgeries= P(k success in n trials)= P (k successes in n trials) =(n¦k) p^k q^(n-k), where:
n = the number of trials = 20
k = the number of successes = 0.7
n – k = number of failures = 0.3
To find the probability that the surgery is successful for at least 5 patients, you can use the binomial probability formula.
The probability of success (p) is 0.70, and the probability of failure (q) is 1 - p, which is 0.30.
To calculate the probability that the surgery is successful for exactly 5 patients (P(x=5)), you can use the formula P(x=k) = (nCk) * (p^k) * (q^(n-k)). Here, n is the number of patients (7), and k is the number of successful surgeries (5).
P(x=5) = (7C5) * (0.70^5) * (0.30^(7-5))
To calculate the probability that the surgery is successful for exactly 6 patients (P(x=6)), you can use the same formula with k = 6.
P(x=6) = (7C6) * (0.70^6) * (0.30^(7-6))
To calculate the probability that the surgery is successful for exactly 7 patients (P(x=7)), you can use the same formula with k = 7.
P(x=7) = (7C7) * (0.70^7) * (0.30^(7-7))
Then, you can sum up these three probabilities to get the probability that the surgery is successful for at least 5 patients:
P(x≥5) = P(x=5) + P(x=6) + P(x=7)
To find the expected number of successes out of 20 surgeries, you can use the expected value formula.
Expected value (E) = n * p
Here, n is the number of surgeries (20), and p is the probability of success (0.70).
Expected value (E) = 20 * 0.70
Keep in mind that the expected number of successes is a measure of what we expect to happen over multiple trials, but it does not guarantee that the exact number of successes will occur.