To find the probability that exactly five out of the next seven patients will survive the delicate heart operation, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
where: P(X = k) is the probability of getting exactly k successes (survivals in this case), C(n, k) is the number of combinations of n items taken k at a time (n choose k), p is the probability of a single success (the probability that a patient recovers, which is 0.9 in this case), and n is the total number of trials (the number of patients in this case).
In this scenario, n = 7 (total number of patients) and p = 0.9 (probability of a patient surviving).
Now, let's calculate the probability of exactly five patients surviving (k = 5):
P(X = 5) = C(7, 5) * 0.9^5 * (1 - 0.9)^(7 - 5)
C(7, 5) is the number of combinations of 7 items taken 5 at a time, which is given by the formula:
C(7, 5) = 7! / (5! * (7 - 5)!) = 7! / (5! * 2!) = 21
Substitute this value into the binomial probability formula:
P(X = 5) = 21 * 0.9^5 * (1 - 0.9)^2
Now, calculate the probability:
P(X = 5) = 21 * 0.9^5 * 0.1^2 ≈ 0.00263
So, the probability that exactly five out of the next seven patients having this delicate heart operation will survive is approximately 0.00263 or 0.263%.