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 20% of the incoming calls involve fax messages, and consider a sample of 20 incoming calls. (Round your answers to three decimal places.) (a) What is the probability that at most 6 of the calls involve a fax message? (b) What is the probability that exactly 6 of the calls involve a fax message? (c) What is the probability that at least 6 of the calls involve a fax message? (d) What is the probability that more than 6 of the calls involve a fax message?

Explanation / Answer

p = 0.2

n = 20

P(X = x) = 20Cx * 0.2x * (1 - 0.2)20-x

a) P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6)

                   = 20C0 * 0.20 * 0.820 + 20C1 * 0.21 * 0.819 + 20C2 * 0.22 * 0.818 + 20C3 * 0.23 * 0.817 + 20C4 * 0.24 * 0.816 + 20C5 * 0.25 * 0.815 + 20C6 * 0.26 * 0.814

                   = 0.913

b) P(X = 6) = 20C6 * 0.26 * 0.814 = 0.109

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

                  = 1 - [P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)]

                  = 1 - [20C0 * 0.20 * 0.820 + 20C1 * 0.21 * 0.819 + 20C2 * 0.22 * 0.818 + 20C3 * 0.23 * 0.817 + 20C4 * 0.24 * 0.816 + 20C5 * 0.25 * 0.815 ]

                  = 1 - 0.804

                  = 0.196

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

                   = 1 - 0.913

                   = 0.087

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