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 \(\boxed{0.0827}\).