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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 7 of the calls involve a fax message?

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Solution:

Total no. of incoming calls, n = 20

Probability of incoming calls with fax messages, p = 20% = 0.20

q = (1 - p) = 0.80

(a) Now, let 'r' be the no. of incoming calls with fax messages, then by Binomial distribution of probability mass function:

P(X = r) = [tex]_{r}^{n}\textrm{C} p^{r}q^{n - r}[/tex]                (1)

P(X ≤ 7) = [tex]_{0}^{20}\textrm{C} (0.20)^{0}(0.80)^{20} +...... + _{7}^{20}\textrm{C} (0.20)^{7}(0.80)^{13}[/tex]

P(X ≤ 7) = 0.0115 +........+ 0.0545

Total no. of incoming calls, n = 20

Probability of incoming calls with fax messages, p = 20% = 0.20

q = (1 - p) = 0.80

(a) Now, let 'r' be the no. of incoming calls with fax messages, then by Binomial distribution of probability mass function:

P(X = r) = [tex]_{r}^{n}\textrm{C} p^{r}q^{n - r}[/tex]                (1)

P(X ≤ 7) = [tex]_{0}^{20}\textrm{C} (0.20)^{0}(0.80)^{20} +...... + _{7}^{20}\textrm{C} (0.20)^{7}(0.80)^{13}[/tex]

P(X ≤ 7) = 0.0115 + 0.0545

P(X ≤ 7) = 0.9689

probability that atmost 7 of the calls are with fax is 0.9689

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