A new antibiotic is effective for 85% of infections. The antibiotic is given to 40 patients.
a) Verify that this is a binomial setting and write it in notation.
b) Describe what P(X=35) means in context.
c) What is the probability that the antibiotic will work in at least 35 of the 40 patients?
d) What is the probability that the antibiotic will work in less than half of the patients?
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a) It is assumed that the administering of the new antibiotic to each patient forms a trial. There are therefore 40 trials, each with the same probability of success (0.850). It is further assumed that the trials are independent of one another and carried out under identical conditions. On the above basis we would have a binomial setting, with a discrete random variable X counting the number of successes in the 40 trials.
The general notation is: X∼Bin(40,0.85)
b) P(X=35) would describe the probability of the antibiotic being effective for 35 people.
c) This describes P(X is greater than or equal to 35). Thus you must find:
P(X is greater than or equal to 35) = P(X=35) + P(X=36) + P(X=37) + P(X=38) + P(X=39) + P(X=40) = (40/35)(0.85)^35(0.15)^5 +…+ (4040)(0.85)^40(0.15)^0.
d) Half the patients is 20. So less than 20 is 19. Hence, here it describes P(X<20)=P(X is lesser than or equal to 19).
that's what i have so far but i dont know how to find C and D