A medication for treating a disease is effective on any give

A medication for treating a disease is effective on any given day with probability P. If the medication is taken for 10 days, and effectiveness is independent from day to day, calculate:

(A) the probability that the medication is effective on all days

(B) the probability that the medication fails on at least 1 of the 10 days

(C) the values of P for which the medication is more likely to be effective on all 10 days than to fail on at least 1 day.

Solution

This is a binomial distribution with n = 10 and p = P

P(x) = 10Cx*(P)^x*(1-P)^(10-x)

A.) P(effective on all days) = P(10) = (P)^10

B.) P(fails on atleast 1 day) = 1 - P(effective on all days) = 1 - (P)^10

C.) (P)^10 > 1 - (P)^10

2*(P)^10 > 1

(P)^10 > 0.5

10*ln(P) > ln(0.5)

So P > 0.93303

So values of P between 0.93303 and 1