A new medicine is developed against a disease to improve the incumbent treatment that is effective in 50% of patients. The expectation is that there is a 2/3 chance that the new treatment will be effective on 60% of patients and a 1/3 chance that the new treatment will be as good as the older one (effective on 50% of patients). The new treatment was given to 20 random patients in the first field trial and was found effective for 15.
With the above information, what is the probability that the new treatment is better than the existing one?
The objective is to find the probability that the new medicine (N) is better, given it was effective on 15 patients in the trial. The notation for this is P(N|X=15) and using Bayes’ theorem,
P(N|X=15) = P(X=15|N) x P(N) / [P(X=15|N) x P(N) + P(X=15|Nc) x P(Nc)]
P(X=15|N) = Probability of getting 15 successes given the probability is 0.6 = 20C15 x (0.6)15 x (0.4)5 (Binomial theorem)
P(X=15|Nc) = Probability of getting 15 successes given the probability is 0.5 = 20C15 x (0.5)15 x (0.5)5 (Binomial theorem)
P(N) = 2/3
P(Nc) = 1/3
dbinom(15, 20, prob = 0.6)*(2/3) / (dbinom(15, 20, prob = 0.6)*(2/3) + dbinom(15, 20, prob = 0.5)*(1/3))0.91
