Question

A particular telephone number is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages, and consider a sample of 20 incoming calls.

a) What is the probability that at least 6 of the calls involve a fax message?

b) What is the probability that exactly 6 of the calls involve a fax message?

c) If calls are monitored one by one, what is the probability that at most eighteen must be selected to find five that do not involve fax messages?

Answer 1

a)

here this is binomial with parameter n=20 and p=0.25

probability that at least 6 of the calls involve a fax message:

P(X>=6)=1-P(X<=5)=

1-∑x=05   (nCx)px(q)(n-x) =

0.3828

(Note:L

if using ti-84 use command :1-binomcdf(20,0.25,5)

if using excel use command :1-binomdist(5,20,0.25,true)

b) probability that exactly 6 of the calls involve a fax message:

P(X=6)=

(20C6)0.256(1−0.25)(20-6)    =

0.1686

c)

here this follows negative binomial distribution with parameter r =5 and p=0.75

P(at most eighteen must be selected to find five that do not involve fax messages)

=1-P(till 18 there are at most 4 that do not invlove fax message)

P(X<=18)=1-P(X>=19)=

1-∑a=04     (xCa)0.25a(1−0.25)(x-a)    =

0.9999961