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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?


(b) What is the probability that exactly 7 of the calls involve a fax message?


(c) What is the probability that at least 7 of the calls involve a fax message?


(d) What is the probability that more than 7 of the calls involve a fax message?

You may need to use the appropriate table in the Appendix of Tables to answer this question.

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

a)
P( X < = 7) = P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 20 7 ) * 0.2^7 * ( 1- 0.2 ) ^13 + ( 20 6 ) * 0.2^6 * ( 1- 0.2 ) ^14 + ( 20 5 ) * 0.2^5 * ( 1- 0.2 ) ^15 + ( 20 4 ) * 0.2^4 * ( 1- 0.2 ) ^16 + ( 20 3 ) * 0.2^3 * ( 1- 0.2 ) ^17 + ( 20 2 ) * 0.2^2 * ( 1- 0.2 ) ^18 + ( 20 1 ) * 0.2^1 * ( 1- 0.2 ) ^19 + ( 20 0 ) * 0.2^0 * ( 1- 0.2 ) ^20   
= 0.9679

b)
P( X = 7 ) = ( 20 7 ) * ( 0.2^7) * ( 1 - 0.2 )^13
= 0.0545

c)
P( X < 7) = P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= ( 20 6 ) * 0.2^6 * ( 1- 0.2 ) ^14 + ( 20 5 ) * 0.2^5 * ( 1- 0.2 ) ^15 + ( 20 4 ) * 0.2^4 * ( 1- 0.2 ) ^16 + ( 20 3 ) * 0.2^3 * ( 1- 0.2 ) ^17 + ( 20 2 ) * 0.2^2 * ( 1- 0.2 ) ^18 + ( 20 1 ) * 0.2^1 * ( 1- 0.2 ) ^19 + ( 20 0 ) * 0.2^0 * ( 1- 0.2 ) ^20   
= 0.9133
P( X > = 7 ) = 1 - P( X < 7) = 0.0867

d)
P( More than 7) = P( X > 7) = 1 - P ( X <=7) = 1 -0.9679 = 0.0321