120 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 1 a. At most 6 of the calls involve a fax message? 2 b. Exactly 6 of the calls involve a fax message? 3 c. At least 6 of the calls involve a fax message? 4 d. More than 6 of the calls involve a fax message? 5 e. The number of calls among the 25 that involve a fax transmission

Answer:a)0.5611b)0.1828c)0.6217d)0.4389e)6.25Step-by-step explanation:Let's define the following event :X : ''Calls involving a fax message''The random variable X can be modeled as a Binomial random variable.N ~ Bi(n,p)N ~ Bi(25,0.25)Where n is the sample and p is the success probabilityLet's also define nCr as the combinatorial number :[tex]nCr=\frac{n!}{r!(n-r)!}[/tex]The probability function for X is :[tex]P(X=k)=nCk.p^{k}.(1-p)^{n-k}[/tex]Where k is the number of successa) [tex]P(X\leq 6)[/tex][tex]P(X\leq 6)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)+P(X=5)+P(X=6)[/tex][tex]P(X\leq 6)=0.5611[/tex]b)[tex]P(X=6)=25C6.(0.25)^{6}.(1-0.25)^{19}=0.1828[/tex]c) [tex]P(X\geq 6)=1-P(X<6)=1-[P(X\leq 6)-P(X=6)]=1-(0.5611-0.1828)\\P(X\geq 6)=1-0.3783=0.6217[/tex]d)[tex]P(X>6)=1-P(X\leq 6)=1-0.5611=0.4389[/tex]e) They are asking us about the expected value for random variable X In a Binomial random variable :E(X) = μ = n.p[tex]E(X)=25(0.25)=6.25[/tex]