Question

Conce
Unknown to a medical researcher, 9 out of 20 patients have a heart problem that will result in death if they receive the test drug. 8 patients are randomly selected to
receive the drug and the rest receive a placebo. What is the probability that exactly 6 patients will die? Express your answer as a fraction or a decimal number rounded to
four decimal places.

Answer 1

Answer:

0.0367 = 3.67% probability that exactly 6 patients will die

Step-by-step explanation:

The patients are chosen without replacement, which means that the hypergeometric distribution is used to solve this question.

Hypergeometric distribution:

The probability of x sucesses is given by the following formula:

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}$

In which:

x is the number of sucesses.

N is the size of the population.

n is the size of the sample.

k is the total number of desired outcomes.

Combinations formula:

$C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

In this question, we have that:

20 patients means that $N = 20$

Sample of 8 means that $n = 8$

9 have the heart problem, so $k = 9$

What is the probability that exactly 6 patients will die?

This is P(X = 6).

$P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}$

$P(X = 6) = h(6,20,8,9) = \frac{C_{9,6}*C_{11,2}}{C_{20,8}} = 0.0367$

0.0367 = 3.67% probability that exactly 6 patients will die