Software filters rely heavily on ""blacklists"" (lists of known ""phishing"" URLs) to detect fraudulent e-mails. But such filters typically catch only 20 percent of phishing URLs. Jason receives 16 phishing e-mails. (a) What is the expected number that would be caught by such a filter? (Round your answer to 1 decimal place.) Expected number (b) What is the chance that such a filter would detect none of them? (Round your answer to 5 decimal places.) Probability

a)  

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

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

The expected value is given by this formula:

b)

Step-by-step explanation:

Previous concepts  

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".  

Part a

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

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

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

The expected value is given by this formula:

Part b

For this case we want this probability:

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

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

And using this function we got:

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is