An experimental drug has been shown to be 70% effective in eliminating symptoms

Question

An experimental drug has been shown to be 70% effective in eliminating symptoms of allergies in animal studies. A small human study involving 9 participants is conducted. What is the probability that the drug is effective on at least 5 of the participants?

Individuals exhibiting a certain set of symptoms are screened for a viral infection. Suppose that: (i) the screening test results in a positive diagnosis for 75% of individuals who really do have the infection; (ii) the screening test results in a negative diagnosis for 75% of individuals who really do not have the infection; and (iii) 10% of individuals exhibiting the set of symptoms have the infection. Let + denote the event that a randomly selected individual exhibiting the set of symptoms is diagnosed positively, and let D denote the event that this individual has the disease.

What is the Pr(+|D)?

What is the Pr(+| )?

What is the Pr(D)?

What is the Pr(D|+)?

Explanation / Answer

drug is effective on at least 5 of the participants = P(X=5) + P(X=6) + P(X=7) + P(X=8) +P(X=9)

= 9C5 * 0.75 * 0.34 + 9C6 * 0.76 * 0.33 + 9C7 * 0.77 * 0.32 + 9C8 * 0.78 * 0.31 + 9C9 * 0.79 * 0.30

=

total = 0.901

let probability of individual having symptoms of infection is = x

then probability of individual not having symptoms of infection is = 1 - x

+ denote the event that a randomly selected individual exhibiting the set of symptoms is diagnosed positively

P(+) = 0.75 * P(D) + 0.25 * (1-x) = 0.5 * x + 0.25

D denote the event that this individual has the disease = P(D)

probability of event that the individual does not have disease but showing symptom = 1 - P(D)

P(D) = 0.1

P(+) = has disease and diagnosed positibely + does not have disease but diagnosed positively = diagnosed positively irrespective of having the disease or not

= 0.1 * 0.75 + 0.9 * (1-0.75) = 0.075+0.225 = 0.3

P(+|D) = has disease and diagnosed properly / has disease = 0.1*0.75 / 0.1 = 0.75

P(D|+) = has disease and diagnosed properly / diagnosed positively = 1*0.75 / 0.3 = 0.25

X 9CX 0.7X 0.3(9-X) 9CX * 0.7X * 0.3(9-X) 5 126 0.16807 0.0081 0.172 6 84 0.117649 0.027 0.267 7 36 0.082354 0.09 0.267 8 9 0.057648 0.3 0.156 9 1 0.040354 1 0.040

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