Problem 3 Suppose 12 of emails received to a particular acco

Problem 3. Suppose 12% of e-mails received to a particular account are classified as spam. Suppose e-mails begin arriving at some point in time. (a) On average, how many of the first 100 e-mails are spam? (b) What is the standard deviation of the number of spam e-mails among the first 100 e-mails? (c) What is the probability that exactly 2 of the first 20 e-mails are spam? (d) Determine the probability that less than 3 of the first 30 e-mails are spam (e) Determine the probability that a least 2 of the first 3 e-mails are spam. (f) Suppose a user spends 5 seconds reading every spam e-mail and 30 seconds reading every non-spam e-mail. What is the expected time spent reading the next 100 received e-mails? (hint: Can you represent total lime as a linear function of X, the number of spam e-mails out of the next 100 received e-mails?)

Solution

p =0.12

E(x) = p * n = 0.12 * 100 = 12 from first 100 are spam

2.

sd = sqrt(n*p*q) = sqrt(100*0.12 * 0.88) = 3.24

3.

P(x=2) = 20C2 * 0.12^2 * 0.88^18

P(x=2) = 0.274

d.

P(x<3) = P(x=0) + P(x=1) + P(x=2)

P(x<3) = 30C0 * 0.12^0 * 0.88^30 + 30C1 0.12^1 0.88^29 + 30C2 0.12^2 ).88^28

P(x<3) = 0.284

e.

P(x>=2) = P(x=2) + P(x=3)

P(x>=2) = 3C2 * 0.12^2 * 0.88^1 + 3C3 * 0.12^3 0.88^0

P(x>=2) = 0.0397