A particular telephone number is used to receive both voice calls and fax messag

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 25 incoming calls. (Round your answers to three decimal places.) (a) What is the probability that at most 8 of the calls involve a fax message? (b) What is the probability that exactly 8 of the calls involve a fax message? (c) What is the probability that at least 8 of the calls involve a fax message? (d) What is the probability that more than 8 of the calls involve a fax message? You may need to use the appropriate table in the Appendix of Tables to answer this question.

Explanation / Answer

Solution:

Given,
Let x denote the calls involve a fax message
n = 25
p = 0.25
The probability mass function of the bynomial distribution is,
P(X = x) = nCx (p)^x (1-p)^n-x

a) The probability that at most 8 of the calls involve a fax message is,
P(X 8) = P(X = 0) +P(X =1) +....+ P(X = 8)
= 25C0 (0.25)^0 (1-0.25)^25-0 +......+ 25C8 (0.25)^8 (1-0.25)^25-8
= 0.0007525 + 0.0062712 + 0.0250848 + 0.0641056 + 0.1175268 + 0.1645376 + 0.1828195+ 0.1654082 + 0.1240561
= 0.8506
b) The probability that exactly 8 of the calls involve a fax message is,
P(X = 8) = 25C8 (0.25)^8 (1-0.25)^25-8
= 0.1240561

c) The probability that at least 8 of the calls involve a fax message is,
P(X 8) = P(X = 0) +P(X =1) +....+ P(X = 7)
= 1- (25C0 (0.25)^0 (1- 0.25)^25-0 +....+ 25C7 (0.25)^7 (1-0.25)^25-7)
= 1- (0.0007525 + 0.0062712 + 0.0250848 + 0.0641056 + 0.1175268 + 0.1645376 + 0.1828195+ 0.1654082)
= 0.2735
d) The probability that more then 8 of the calls involve a fax message is,
P(X > 8) = P(X = 9) +....+ P(X = 25)
= 25C9 (0.25)^9 (1-0.25)^25-9 +......+ 25C25 (0.25)^25 (1-0.25)^25-25
= 0.0781094 + 0.0416584 + 0.0189356 + 0.0073638 + 0.0024546 + 0.0007013 + 0.0001714 + 0.0000357 + 0.0000063 + 0.0000009 + 0.0000001 + 0.0000000 + 0.0000000 + 0.0000000+ 0.0000000+ 0.0000000+ 0.0000000
= 0.1494

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