1. A student uses two email accounts. Spam is sent regularly to both account: 30% of spam is sent to the UTEP account and 70% to the gmail account. UTEP filters out 80% of the spam, gmail filters out 90% of the spam. Overall, what percentage of the original spam reaches a student?
2. What is the probability that a spyware that tries 100 random guesses will guess a password consisting of 3 digits and one lower-case letter?
3. A student who studied for the quiz answers correctly with probability 90%, while a student who did not study answers correctly with probability 10%. We know that 80% of the students studied for the quiz. If a student answered correctly, what is the probability that he did not study?
4. A computer is connected to three different printers. The first and the second printers more reliable; for each of them, the probability of failure is 10%. For the third one, the probability of failure is 40%. What is the probability that at least one of the printers is available?
5. In a remote CS Department, among 10 faculty, 5 are good teachers and 5 are not so good teachers, and a new student does not know which are good and which are not. What is the probability that, when a student signs for 3 classes, all his professors are good?
6-7. Two random variables X and Y describe whether the two printers are working: X = 1 is the first printer works and X = 0 if it does not; Y = 1 is the second printer works and Y = 0 if it doesn't. The joint distribution is P(0,0) = 0.2, P(0,1) = 0.4, P(1,0) = 0.3, P(1,1) = 0.1, Find the distribution for X, Y, and for the auxiliary variable U = X + Y - X * Y (describing when one of the printers work). Are variables X and Y independent? Compute the expected value E(U) and the variance V(U).
8. Suppose now that the variables X and Y corresponding to two printers are independent; the variable X has value 1 with probability 80% and the variable Y has value 1 with probability 60%. What is then the expected value of the auxiliary variable U = X + Y - X * Y?
9. A computer virus attacks a folder consisting of 50 executable files. Each files is affected with the probability 0.01. What is the probability that more than 3 files will be affected by the virus?