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Unknown to a medical researcher, 6 out of 25 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.

Respuesta :

Using the hypergeometric distribution, it is found that there is a 0.0002 = 0.02% probability that exactly 6 patients will die.

What is the hypergeometric distribution formula?

The formula is:

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The parameters are:

  • x is the number of successes.
  • N is the size of the population.
  • n is the size of the sample.
  • k is the total number of desired outcomes.

The values of the parameters for this problem are:

N = 25, k = 6, n = 8.

The probability that exactly 6 patients will die is P(X = 6), hence:

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}C_{N-k,n-x}}{C_{N,n}}[/tex]

[tex]P(X = 6) = h(6,25,8,6) = \frac{C_{6,6}C_{19,2}}{C_{25,8}} = 0.0002[/tex]

0.0002 = 0.02% probability that exactly 6 patients will die.

More can be learned about the hypergeometric distribution at https://brainly.com/question/24826394

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