Question

Twocombinationdrugtherapies(TreatmentAandTreatmentB)have been developed for eradicating Helicobacter pylori in human patients. The effectiveness of these treatments depends on whether or not the patient is resistant to the chemical compound Metronidazole, but apatient’s resistance status is not routinely determined before beginningtreatment. Treatment A successfully eradicates Helicobacter pylori in 92% of resistant patients and 87% of non-resistant patients. The corresponding proportions for Treatment B are 75% and 95%.

(i) Denote by θ (0 < θ < 1) the proportion of the affected population that is resistant. If a patient from this population is unsuccessfully treated with Treatment B, write down an expression for the probability that the patient is resistant.

(ii) For what values of θ would a greater proportion of patients from this population be successfully treated by Treatment B than by Treatment A?

(iii) Suppose that θ = 0.2. If 20 patients, selected at random from the affected population, are independently treated with Treatment B, find the probability that at least 18 of them will be treated successfully.

Answer 1

Solution

Let R represent the event that the patient is resistant; A represent the event that Treatment A is successful in treating the patient and B represent the event that Treatment B is successful treating the patient. Then, trivially,

R^C represents the event that the patient is non-resistant; A^C represents the event that Treatment A is unsuccessful in treating the patient and B^C represents the event that Treatment B is unsuccessful in treating the patient.

Back-up Theory

If A and B are two events such that probability of B is influenced by occurrence or otherwise of A, then

Conditional Probability of B given A, denoted by P(B/A) = P(B ∩ A)/P(A)……………………….(1)

P(B) = {P(B/A) x P(A)} + {P(B/A^C) x P(A^C)}……………………………………………………..….(2)

P(A/B) = P(B/A) x {P(A)/P(B)}..………………………….………………………..………………….(3)

Now to work out the solution,

With the above terminology, the given data, in probability language would be:

P(R) = θ and hence P(R^C) = (1 – θ) ……………………….………….....................……………..(4)

P(A/R) = 0.92, P(A/R^C) = 0.87 and hence P(A^C/R) = 0.08, P(A^C/R^C) = 0.13 …. ....................(5)

P(B/R) = 0.75, P(B/R^C) = 0.95 and hence P(B^C/R) = 0.25, P(B^C/R^C) = 0.15 …. ....................(6)

Now, vide (2),

P(A) = (0.92θ) + {0.87(1 - θ)} = 0.05 θ + 0.87 …………………………………...........................(7)

P(B) = (0.75θ) + {0.95(1 - θ)} = 0.95 - 0.2θ ….…………………………………..........................(8)

(7) => P(A^C) = 0.13 - 0.05 θ ……………………………………………………….........................(9)

(8) => P(B^C) = 0.2θ + 0.05 ……………………………………………………….........................(10)

Part (i)

If a patient from this population is unsuccessfully treated with Treatment B, probability that the patient is resistant = P(R/B^C)

= {P(B^C/R) x P(R)}/{P(B^C)} [vide (2)]

= 0.25θ/(0.2θ + 0.05) [vide (6), (4) and (10)] ANSWER

Part (ii)

Given condition => 0.95 - 0.2θ > 0.05 θ + 0.87 [vide (8) and (7)]

Or, 0.2θ < 0.08

Or, θ < 0.32 ANSWER

Part (iii)

Given θ = 0.2, P(B) = 0.91 [vide (8)]

So, if X = number of patients out of 20, who are treated successfully by Treatment B, then X ~ B(20, 0.91) and hence, the required probability

= P(X ≥ 18)

= 1 - P(X ≤ 17)

= 1 – 0.26657 [using Excel Function: BINOMDIST(Number_s:Trials:Probability_s:Cumulative)]

= 0.73343 ANSWER

DONE