Question

There is an old drug for a certain disease. The cure rate of the old drug (the proportion of patients cured) is 0.8.

Pharma Co has developed a new drug for this disease. The company is conducting a small field trial (trial with real patients) of the new drug and the old drug. Assume that patient outcomes are independent. Also, assume the probability that any one patient will be cured when they take the drug is p.  For the old drug, p = 0.8. The random variable X is the number of patients who are cured when the drug is given to n patients.

A. The old drug is given to 16 randomly-chosen patients. What is the expected number of patients who will be cured?  One decimal.

B. The old drug is given to 16 randomly-chosen patients. What is the probability that all 16 will be cured?  Four decimals.

C. Pharma Co thinks the answer to the previous question is too big. They want the probability that all of the patients will be cured to be less than 0.01. (We mean “all of the patients who get the old drug in the field trial”.) What is the smallest number of patients that Pharma Co must give the old drug to? HINT: You can do this either by trial and error (increasing n above 16), or by setting up an inequality with n as an unknown, and taking the natural logarithm of it to solve for n.  Integer.

D. Alternatively, suppose Pharma Co uses a sample size of 35. The old drug is given to 35 randomly-chosen patients. What is the probability that all 35 will be cured?  Four decimals.

E. Pharma Co thinks the answer to the previous question is too small! They want P(X  a)  0.01 , and they want the smallest value of “a” where P(X  a)  0.01. In other words, they don’t necessarily want to use a = 35. What about a = 34? Is P(X  34)  0.01? Is P(X  33)  0.01. Find the smallest value of “a” where P(X  a)  0.01. NOTE: We are not changing n. We are still giving the old drug to 35 patients. We are just calculating P(X  a), when n = 35 and p = 0.8, for different values of “a”, to find the value of “a” that we want.  Integer.

F. For the new drug, p = 0.95. Go back to your answer to the third question in this set (“smallest number of patients …”). Suppose Pharma Co gives the new drug to this many patients. What is the probability they all will be cured?  Four decimals.

G. Go back to your answer to the question E in this set (“find the smallest value of a where …”). Use the value of “a” that you calculated in that question. What is P(X ≥≥ a) if the new drug (p = 0.95) is given to 35 patients?  Four decimals

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}, ………….............................................................................………..(1)

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

Mean (average) of X = E(X) = µ = np…....................................................................................….....……..(2)

Now to work out the solution,

As given,

X = number of patients who are cured when the drug is given to n patients. Then,

X ~ B(n, p), ……………………………………………………………………………....................................…(3)

where p = probability the drug would cure which is also equal to the cure rate of the drug.

Part (A)

Given,

‘The old drug is given to 16 randomly-chosen patients and the cure rate of the old drug is 0.8.’, X ~ B(16, 0.8)

So, vide (2),

expected number of patients who will be cured = 16 x 0.8 = 12.8. Answer 1

Part (B)

Here again, X ~ B(16, 0.8). So, required probability is:

P(X = 16)

= 0.0281 [vide (1a)] Answer 2

Part (C)

Here, we want to find n such that P(X = n) < 0.01

i.e., (ⁿC_n)(0.8ⁿ)(0.2)⁰ < 0.01

i.e., (0.8ⁿ)< 0.01

Or, taking natural logarithm, nln0.8 < ln0.01

i.e., - 0.2231n < - 4.6051

Or, n > (- 4.6051)/( - 0.2231) [Note that this step involves division by negative number which reverses the inequality sign.]

So, n > 20.63.

Thus,

smallest number of patients that Pharma Co must give the old drug to is 21 Answer 3

Part (D)

Here, n = 35 and X ~ B(35, 0.8)

So, P(X = 35)

= 0.0004 [vide (1a)] Answer 4

Answer

Part (E)

Here, we want to find a such that P(X = a) = 0.01, with X ~ B(35, 0.8)

i.e., (³⁵C_a)(0.8^a)(0.2)^{35 - a} = 0.01

The above can be solved only by trial and error as shown below: [we use (1a)]

a	P(X = a)
35	0.00040565
34	0.00354942
33	0.01508504
32	0.04148387

Thus,

smallest value for a is 33 Answer 5

DONE