A particular telephone number is used to receive both voice calls and fax message calls. Suppose that 25% of the incoming calls are fax message calls. Among the next 25 calls,find the probability that the number of calls that involve a fax message exceeds the expected number by more than 2 standard deviations.

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Solution to the problem

Let X the random variable of interest, on this case we now that:

[tex]X \sim Binom(n=25, p=0.25)[/tex]

The probability mass function for the Binomial distribution is given as:

[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]

Where (nCx) means combinatory and it's given by this formula:

[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]

The expected value is given by:

[tex] E(x) = np =25*0.25=6.25[/tex]

And the deviation given by:

[tex] Sd(X) = \sqrt{25*0.25*(1-0.25)}= 2.165[/tex]

And we want to find this probability:

[tex] P(X > 6.25 +2*2.165) = P(X>10.58)[/tex]

And we can find this probability like this:

[tex] P(X >10) [/tex]

And we can use the complement rule given by:

[tex]P(X >10)= 1-P(X\leq 10)[/tex]

And we can ue the following excel code :

=1-BINOM.DIST(10,25,0.25,TRUE)

And we got 0.0297