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. What is the probability that a.? ?At most 6 of the calls involve a fax message? b.? ?Exactly 6 of the calls involve a fax message? c.? ?At least 6 of the calls involve a fax message? d.? ?More than 6 of the calls involve a fax message?

Answer: Step1: Given, 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. Here, n = 25 and p = 25% = 0.25 Let x follows binomial distribution with parameters (n,p). i.e, X ~ B(n,p) x P(x;p,n) = (c )x (1 p) n x i = 0 Step2: a). To find the probability that at most 6 of the calls involve a fax message. Let x follows number of calls involving a fax message. 6 25 x 25x P(x 6) = (c ) (x.25) (1 0.25) i =0 = c( )5 (0.25) (1 0.25)250 + (1 )(0.25) (1 0.25) 251+ (c2)(0.25) (1 0.25) 252 0 + c 25 (0.25) (1 0.25)253+ c 25 (0.25) (1 0.25)254 + c 25 (0.25) (1 0.25)255 ( )3 ( )4 ( )5 + ( )25 (0.25) (1 0.25)256 6 = 0.0008 + 0.0063 + 0.0251+ 0.0641 + 0.1175 + 0.1645 + 0.1828 = 0.5611 Therefore,the probability...