Question

A cardiac scientist wishes to conduct a study on humans in two stages. Both stages involve experimental surgeries, but only some of the subjects will continue to the second stage, depending on their outcome from the first stage. The probability of a randomly selected subject from stage 1 continuing to the second stage is .35.

Both stages are very expensive and the scientist must balance starting with enough subjects to complete the entire study, against the cost of each additional subject in the study. The scientist decided the study needs at least 10 subjects to complete the entire study, but they can financially afford no more than 25 to complete the entire study. The scientist believes that starting the study with 50 subjects will put them in this 10 – 25 range at the end.

What is the actual probability that the study will finish with 10 to 25 subjects assuming the study begins with 50? Show all your work for credit.

A cardiac scientist wishes to conduct a study on humans in two stages. Both stages involve experimental surgeries, but only some of the subjects will continue to the second stage, depending on their outcome from the first stage. The probability of a randomly selected subject from stage 1 continuing to the second stage is .35.

Both stages are very expensive and the scientist must balance starting with enough subjects to complete the entire study, against the cost of each additional subject in the study. The scientist decided the study needs at least 10 subjects to complete the entire study, but they can financially afford no more than 25 to complete the entire study. The scientist believes that starting the study with 50 subjects will put them in this 10 – 25 range at the end.

What is the actual probability that the study will finish with 10 to 25 subjects assuming the study begins with 50? Show all your work for credit.

Answer 1

Solution

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and

p = probability of one success, then, probability mass function (pmf) of X is given by

p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}, x = 0, 1, 2, ……. , n ………….........................................................................………..(1)

[This probability can also be directly obtained using Excel Function: Statistical, BINOMDIST]....................….........……….(1a)

Now to work out the solution,

Let X = number of subjects out of a sample 50 subjects who complete the entire study.

Then, X ~ B(50. 0.35) .............................................................................................................................................................(2)

Probability that the study will finish with 10 to 25 subjects

= P(10 ≤ X ≤ 25)

= P(X ≤ 25) - P(X ≤ 9)

= 0.9900 – 0.0067 [vide (1) and (1a)]

= 0.9833 Answer

DONE