To calculate the probability that exactly five out of the next seven patients will survive the delicate heart operation, we can use the binomial probability formula. The binomial probability formula is given by:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where: P(X = k) is the probability of getting exactly k successes. n is the total number of trials (in this case, the number of patients undergoing the operation). k is the number of successful outcomes we want (in this case, the number of patients surviving the operation). p is the probability of success on each trial (in this case, the probability that a patient will recover, which is 0.9). C(n, k) is the number of ways to choose k successes from n trials, also known as the binomial coefficient, and can be calculated as C(n, k) = n! / (k! * (n - k)!).
Now, let's calculate the probability:
n = 7 (total number of patients undergoing the operation) k = 5 (number of patients we want to survive) p = 0.9 (probability that a patient will recover)
C(7, 5) = 7! / (5! * (7 - 5)!) = 21 (7 choose 5)
P(X = 5) = 21 * (0.9)^5 * (1 - 0.9)^(7 - 5) P(X = 5) = 21 * 0.9^5 * 0.1^2 P(X = 5) = 21 * 0.59049 * 0.01 P(X = 5) = 0.1244109
So, the probability that exactly five out of the next seven patients will survive the delicate heart operation is approximately 0.1244 or 12.44%.