Question

In a large university, 15% of the students are female. If a random sample of twenty students is selected, a. what is the probability that the sample contains exactly four female students? b. what is the probability that the sample will contain no female students? c. what is the probability that the sample will contain exactly twenty female students? d. what is the probability that the sample will contain more than nine female students? e. what is the probability that the sample will contain fewer than five female students? f. what is the expected number of female students?

Answer 1

Here we see that p=0.15 is same for all, n=20 is constant, events are independent and only two outcomes.

Hence all the properties of binomial are satisfied

So we will use binomial distribution to find the required probabilities

a. $P(x=4)=BINOMDIST(4,20,0.15,0)=0.1821$

b. $P(x=0)=BINOMDIST(0,20,0.15,0)=0.0388$

c. $P(x=20)=BINOMDIST(20,20,0.15,0)=0.0000$

d. $P(x>9)=1-P(x\leq9)=1-BINOMDIST(9,20,0.15,1)=1-0.9998=0.0002$

e. $P(x<5)=P(x\leq 4)=0.8298$

f. $E(x)=np=20*0.15=3$