Questions: Coronary bypass surgery: A healthcare agency reported that 51% of people who had coronary bypass surgery in a recent year 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. The probability that exactly 11 of them are over the age of 65 is 0.1040. Part 2 of 4 (b) What is the probability that more than 12 are over the age of 65? Round the answer to four decimal places.

Solution Steps

To find the probability that exactly 11 out of 17 coronary bypass patients are over the age of 65, we use the binomial probability formula:

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

where:

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

Calculating this gives:

\[ P(X = 11) = \binom{17}{11} \cdot (0.51)^{11} \cdot (0.49)^{6} \approx 0.104 \]

Thus, the probability that exactly 11 patients are over the age of 65 is:

\[ \boxed{0.104} \]

To find the probability that more than 12 patients are over the age of 65, we need to calculate the probabilities for \( X = 13, 14, 15, 16, \) and \( 17 \) and sum them up:

\[ P(X > 12) = P(X = 13) + P(X = 14) + P(X = 15) + P(X = 16) + P(X = 17) \]

Calculating each term:

• For \( X = 13 \): \[ P(X = 13) \approx 0.0217 \]

• For \( X = 14 \): \[ P(X = 14) \approx 0.0064 \]

• For \( X = 15 \): \[ P(X = 15) \approx 0.0013 \]

• For \( X = 16 \): \[ P(X = 16) \approx 0.0002 \]

• For \( X = 17 \): \[ P(X = 17) \approx 0.0 \]

Summing these probabilities:

\[ P(X > 12) \approx 0.0217 + 0.0064 + 0.0013 + 0.0002 + 0.0 \approx 0.0296 \]

Thus, the probability that more than 12 patients are over the age of 65 is:

\[ \boxed{0.0296} \]

Final Answer