The probability that patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 5 of the next 7 patients having this operation survive?
C(7,5)(.9)^5(.1)^2
= .124
The survival rate during a risky operation for patients with no other hope of survival is 74%. What is the probability that exactly four of the next five patients survive this operation? (Give your answer correct to three decimal places.)
To find the probability that exactly 5 of the next 7 patients survive, we can use the binomial probability formula.
The formula for the probability of exactly k successes in n trials, given the probability p of success in each trial, is:
P(X = k) = (nCk) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of getting k successes.
n is the total number of trials or patients.
k is the number of successes.
p is the probability of success in each trial or the probability of a patient surviving.
nCk is the number of combinations of n things taken k at a time.
In this case, n = 7, k = 5, and p = 0.9.
Let's calculate the probability using the formula:
P(X = 5) = (7C5) * 0.9^5 * (1 - 0.9)^(7 - 5)
To calculate (nCk), we use the formula:
nCk = n! / (k! * (n - k)!)
So:
(7C5) = 7! / (5! * (7 - 5)!)
= (7 * 6 * 5!) / (5! * 2)
= (7 * 6) / 2
= 42 / 2
= 21
Now we can substitute the values into the formula:
P(X = 5) = (21) * (0.9^5) * (1 - 0.9)^(7 - 5)
= 21 * 0.9^5 * 0.1^2
= 21 * 0.59049 * 0.01
= 0.1244651
Therefore, the probability that exactly 5 of the next 7 patients having this operation survive is approximately 0.1245 or 12.45%.
To find the probability that exactly 5 of the next 7 patients survive, we can use the binomial probability formula. The binomial probability formula calculates the probability of a specific number of successes in a fixed number of trials.
The formula is:
P(x) = (nCx) * (p^x) * (q^(n-x))
Where:
P(x) is the probability of getting exactly x successes
n is the total number of trials (in this case, the number of patients)
x is the number of successes (in this case, the number of patients surviving)
p is the probability of success on a single trial (in this case, the probability that a patient survives)
q is the probability of failure on a single trial (in this case, the probability that a patient does not survive), which is equal to 1 - p
Using the given information:
n = 7
x = 5
p = 0.9
q = 1 - p = 1 - 0.9 = 0.1
Now, let's substitute these values into the formula:
P(5) = (7C5) * (0.9^5) * (0.1^(7-5))
To calculate the combination (7C5), we can use the formula:
nCr = n! / (r! * (n-r)!)
(7C5) = 7! / (5! * (7-5)!)
= 7! / (5! * 2!)
Calculating the factorials:
7! = 7 * 6 * 5!
5! = 5 * 4 * 3 * 2!
2! = 2 * 1
Substituting back into the combination formula:
(7C5) = (7 * 6 * 5!) / (5! * 2!)
= (7 * 6) / 2
= 21
Substituting all the values back into the binomial probability formula:
P(5) = 21 * (0.9^5) * (0.1^2)
Now, we can calculate the probability:
P(5) = 21 * 0.9^5 * 0.1^2
P(5) = 21 * 0.59049 * 0.01
P(5) = 0.1240
Therefore, the probability that exactly 5 of the next 7 patients recover from a delicate heart operation is approximately 0.1240 or 12.40%.