Coronary bypass surgery:a health care agency reported that 49%of people who had coronary bypass surgery in 2008 were over the age of 65.fifteen coronary bypass patients are sampled
Part 1 of 4
(a) What is the probability that exactly 10 of them are over the age of 65?round the answer to four decimal places.
The probability that exactly 10 of them are over the age of 65 is

Step 1 ：This problem is a binomial distribution problem. The binomial distribution model is appropriate for a statistical experiment if the following conditions are met:

Step 2 ：1. The experiment consists of n repeated trials.

Step 3 ：2. Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.

Step 4 ：3. The probability of success, denoted by P, is the same on every trial.

Step 5 ：4. The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

Step 6 ：In this case, we have n=15 (the number of patients), k=10 (the number of successes we want, i.e., the number of patients over the age of 65), and p=0.49 (the probability of success on each trial, i.e., the probability a patient is over the age of 65).

Step 7 ：The formula for the binomial distribution is: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\), where \(C(n, k)\) is the number of combinations of n items taken k at a time.

Step 8 ：Substituting the given values into the formula, we get \(P(X=10) = C(15, 10) * (0.49^{10}) * ((1-0.49)^{15-10})\).

Step 9 ：Calculating the above expression, we get a probability of 0.0827.

Step 10 ：Final Answer: The probability that exactly 10 of them are over the age of 65 is \(\boxed{0.0827}\).