Questions: Coronary bypass surgery: A healthcare agency reported that 51% of people who had coronary bypass surgery in 2008 were over the age of 65. Seventeen coronary bypass patients are sampled. Part 1 of 4 (a) What is the probability that exactly 11 of them are over the age of 65? Round the answer to four decimal places.

Solution Steps

We are tasked with finding the probability that exactly 11 out of 17 coronary bypass patients are over the age of 65, given that the probability of a patient being over the age of 65 is \( p = 0.51 \).

The probability of exactly \( x \) successes in \( n \) trials can be calculated using the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where:

• \( n = 17 \) (the number of trials),
• \( x = 11 \) (the number of successes),
• \( p = 0.51 \) (the probability of success),
• \( q = 1 - p = 0.49 \) (the probability of failure).

Substituting the values into the formula, we have:

\[ P(X = 11) = \binom{17}{11} \cdot (0.51)^{11} \cdot (0.49)^{6} \]

Calculating \( \binom{17}{11} \):

\[ \binom{17}{11} = \frac{17!}{11!(17-11)!} = \frac{17!}{11!6!} = 12376 \]

Now, substituting this into the probability formula:

\[ P(X = 11) = 12376 \cdot (0.51)^{11} \cdot (0.49)^{6} \]

After performing the calculations, we find:

\[ P(X = 11) \approx 0.104 \]

Final Answer