Anew 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?

(a)
The binomial distribution can be used because the current situation satisfies all of the following:
1. The probability of success (p=85%) is known and remains constant during the whole experiment
2. The number of trials (n=40) is known and constant.
3. Each trial is a bernoulli trial (success or failure only)
4. All trials are (assumed) independent of each other.
The probability of x successes is therefore
P(X=x)=C(n,x)(p^x)(1-p)^(n-x)

(b) P(X=35) means the probability of 35 successes out of 40 trials at p=0.85
and
P(X=35)=C(40,35)*0.85^35*0.15^5=658008*0.003386*0.00007594
=0.16918

(c) P(X>=35)=∑ P(X=i) for i=35 to 40
=0.16918+0.13315+0.08157+0.03649+0.01060+0.00150
=0.4325

(d) P(X<20)=&sum; P(X=i) for i=0 to 19
=0.00000003513   (individual probabilities are very small).