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 probaytht more than 8 of the calls involve a fax message?

Explanation / Answer

This is Bin(25,0.25) distribution.

p(x) = nCx px (1-p)n-x

a)

p(x<=8) = p(x=0) + p(x=1) + p(x=2) + p(x=3) + p(x=4) + p(x=5) + p(x=6) + p(x=7) + p(x=8)

= 25C0 p25 (1-p)25-0 +25C1 p25 (1-p)25-1 +25C2 p25 (1-p)25-2 +25C3 p25 (1-p)25-3 +25C4 p25 (1-p)25-4 +

25C5 p25 (1-p)25-5 +25C6p25 (1-p)25-6+25C7 p25 (1-p)25-7 +25C8 p25 (1-p)25-8

= 0.8506

b)

p(x=8) = 25C8 p25 (1-p)25-8

= 0.1241

c)

p(x>=8) = 1-p(x<=7)

= 1 -{ 25C0 p25 (1-p)25-0 +25C1 p25 (1-p)25-1 +25C2 p25 (1-p)25-2 +25C3 p25 (1-p)25-3 +25C4 p25 (1-p)25-4 +

  25C5 p25 (1-p)25-5 +25C6p25 (1-p)25-6+25C7 p25 (1-p)25-7 }

= 1 - 0.7265

= 0.2735

d)

p(x>8) = 1 - p(x<=8)

= 1 - 0.8506

= 0.1494

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