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

0.0367 = 3.67% probability that exactly 6 patients will die

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) = (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) = (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) = (C_(k,x)*C_(N-k,n-x))/(C_(N,n))`

`P(X = 6) = h(6,20,8,9) = (C_(9,6)*C_(11,2))/(C_(20,8)) = 0.0367`

0.0367 = 3.67% probability that exactly 6 patients will die