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Coronary bypass surgery: The Agency for Healthcare Research and Quality reported that
60% of people who had coronary bypass surgery in a recent year were over the age of 65 , Fifteen coronary bypass patients are sampled. Round the answers to at least four decimal places.
Part:
0//4
Part 1 of 4
(a) What is the probability that exactly 10 of them are over the age of 65 ?
The probability that exactly 10 of them are over the age of 65 is
◻

Coronary bypass surgery: The Agency for Healthcare Research and Quality reported that $60 \%$ of people who had coronary bypass surgery in a recent year were over the age of $65$ , Fifteen coronary bypass patients are sampled. Round the answers to at least four decimal places. $\newline$ Part: $0 / 4$ $\newline$ Part $1$ of $4$ $\newline$ (a) What is the probability that exactly $10$ of them are over the age of $65$ ? $\newline$ The probability that exactly $10$ of them are over the age of $65$ is $\square$

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Find probabilities using the binomial distribution

Full solution

Q. Coronary bypass surgery: The Agency for Healthcare Research and Quality reported that

60 \%

of people who had coronary bypass surgery in a recent year were over the age of

65

, Fifteen coronary bypass patients are sampled. Round the answers to at least four decimal places.

\newline

Part:

0 / 4

\newline

Part

1

of

4

\newline

(a) What is the probability that exactly

10

of them are over the age of

65

?

\newline

The probability that exactly

10

of them are over the age of

65

is

\square

Identify values: We're dealing with a binomial probability problem here. The formula for binomial probability is $P(X=k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$ , where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single trial, and $\binom{n}{k}$ is the binomial coefficient.
Calculate binomial coefficient: First, let's identify the values we need: $n = 15$ (the number of patients), $k = 10$ (the number of patients we want to have had the surgery who are over $65$ ), and $p = 0.60$ (the probability that a patient is over $65$ ).
Calculate factorial values: Now, calculate the binomial coefficient $15 \choose 10$ . This is equal to $\frac{15!}{10! \cdot (15-10)!}$ .
Divide factorial values: Calculating the factorial values, we get $15! = 1,307,674,368,000$ , $10! = 3,628,800$ , and $5! = 120$ .
Simplify division: Now, divide $15!$ by $10!$ and $5!$ to get the binomial coefficient: $\binom{15}{10} = \frac{1,307,674,368,000}{(3,628,800 \times 120)}$ .
Simplify division: Now, divide $15!$ by $10!$ and $5!$ to get the binomial coefficient: $\binom{15}{10} = \frac{1,307,674,368,000}{(3,628,800 \times 120)}$ .Simplify the division to get $\binom{15}{10} = 3,003$ .

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