Exam_1

Ω is a sample space. A is a subset of Ω. Find A∩ Ω and AUΩ.

A∩ Ω = A, AUΩ = Ω

2.15. A computer program consists of two blocks written independently by two different programmers. The first block has an error with probability 0.2. The second block has an error with probability 0.3. If the program returns an error, what is the probability that there is an error in both blocks?

0.06

2.9. Successful implementation of a new system is based on three independent modules. Module 1 works properly with probability 0.96. For modules 2 and 3, these probabilities equal 0.95 and 0.90. Compute the probability that at least one of these three modules fails to work properly.

0.1792

Two tickets are drawn at random without replacement from a box containing 100 tickets with different numbers, of which 25 are red, 25 black, 25 purple, and 25 blue. What is the probability the second ticket is black?

0.25

2.3. A new computer virus can enter the system through e-mail or through the internet. There is a 30% chance of receiving this virus through e-mail. There is a 40% chance of receiving it through the internet. Also, the virus enters the system simultaneously through e-mail and the internet with probability 0.15. What is the probability that the virus does not enter the system at all?

0.45

2.2. Suppose that after 10 years of service, 40% of computers have problems with motherboards (MB), 30% have problems with hard drives (HD), and 15% have problems with both MB and HD. What is the probability that a 10-year old computer still has fully functioning MB and HD?

0.55

A probability distribution is given. X 0 1 2 3 4 P(X) 0.2 0.3 0.1 0.15 0.25

0.6

2.7. A system may become infected by some spyware through the internet or e-mail. Seventy percent of the time the spyware arrives via the internet, thirty percent of the time via email. If it enters via the internet, the system detects it immediately with probability 0.6. If via e-mail, it is detected with probability 0.8. What percentage of times is this spyware detected?

0.63

Three computer viruses arrived as an email attachment. Virus A damages the system with probability 0.4. Independently of it, virus B damages the system with probability 0.5. Independently of A and B, virus C damages the system with probability 0.2. What is the probability that the system gets damaged?

0.76

2.13. An important module is tested by three independent teams of inspectors. Each team detects a problem in a defective module with probability 0.8. What is the probability that at least one team of inspectors detects a problem in a defective module?

0.992

Sam is going to assemble a computer by himself. he has choice of chips from two bands, a hard dive from four, memory from three and an accessory bundle from five local stores. How many different ways can Sam order the parts?

1/120

2.1. Out of six computer chips, two are defective. If two chips are randomly chosen for testing (without replacement), compute the probability that both of them are defective. List all the outcomes in the sample space.

A=(G,G,G,G,B,B) =>2 bad out of 6 chips =2/6 B=(G,G,G,G,B)=> 1 bad out of 5 chips =1/5 (AnB)=(2/6)(1/5)=1/15

Ω = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A = {1, 2, 3, 4, 5}, B = {6, 7, 8, 9, 10}, C = {5, 6}. Find A∩B and AUC.

A∩B = empty set , AUC = {1, 2, 3, 4, 5, 6}

3.7. The number of home runs scored by a certain team in one baseball game is a random variable with the distribution The team plays 2 games. The number of home runs scored in one game is independent of the number of home runs in the other game. Let Y be the total number of home runs. Find E(Y ) and Var(Y ).

E(Y)=1.6 var(Y)= 1.12

The following table represents a legitimate probability distribution. X 1 2 3 4 P(X) -0.2 0.3 0.4 0.5

False

A probability distribution is given. X 0 1 2 3 4 P(X) 0.2 0.3 0.1 0.15 0.25

.25

A probability distribution is given. X -1 1 2 P(X) 0.2 0.3 0.5 find cdf

0 x<-1 0.2 -1<_x<1 0.5 1<_x<2 1 x>_2

A fair dice (six-sided) is rolled twice. What is the probability that the sum of the numbers rolled will add up to seven? List the outcomes representing the sum of 7.

samplespace={(1,1),(1,12),(1,3),(1,4),(1,5),(1,6), (2,1),(2,2),(2,3),(2,4),(2,5),(2,6), (3,1),(3,2),(3,3),(3,4),(3,5),(3,6), (4,1),(4,2),(4,3),(4,4),(4,5),(4,6), (5,1),(5,2),(5,3),(5,4),(5,5),(5,6), (6,1),(6,2),(6,3),(6,4),(6,5),(6,6)}= 36 Event A={(1,6),(2,5),(3,4),(4,3),(5,6),(6,1)}=6 p(roll 7)= 1/6

#8. Four cards will be dealt off the top of a well-shuffled deck. There are two options. (i) To win $10 if the first card is club and the second is a diamond and the third is a heart and the fourth is a spade. (ii) To win $10 if the four cards are of four different suits. Compute the probability of winning $10 for each case. Which option is better?

N/A

One ticket will be drawn at random from the box below. Are color and number independent? 1 1 8 1 1 8 In case you do not see the colors, the first three tickets are yellow and the last three are green.

N/A

Which of the following is incorrect?

P(A∩B) = P(A)P(B|A) for any two events.

2.18. A problem on a multiple-choice quiz is answered correctly with probability 0.9 if a student is prepared. An unprepared student guesses between 4 possible answers, so the probability of choosing the right answer is 1/4. Seventy-five percent of students prepare for the quiz. If Mr. X gives a correct answer to this problem, what is the chance that he did not prepare for the quiz?

P(Not prepared)=.25 p(Not prepare| correct)=0.084

The following table represents a legitimate probability distribution. X -1.5 2 2.5 P(X) 0.2 0.4 0.4

True

3.1. A computer virus is trying to corrupt two files. The first file will be corrupted with probability 0.4. Independently of it, the second file will be corrupted with probability 0.3. (a) Compute the probability mass function (pmf) of X, the number of corrupted files. (b) Draw a graph of its cumulative distribution function (cdf).

a) p(x=0)= 0.42 p(x=1)= 0.46 p(x=2)= 0.12 b) graph

2.14. A spyware is trying to break into a system by guessing its password. It does not give up until it tries 1 million different passwords. What is the probability that it will guess the password and break in if by rules, the password must consist of (a) 6 different lower-case letters (b) 6 different letters, some may be upper-case, and it is case-sensitive (c) any 6 letters, upper- or lower-case, and it is case-sensitive (d) any 6 characters including letters and digits

a) 0.0060325147 b) 6.82215x10^-5 c) 5.05801x10^-5 d) 1.76056x10^-5

Two tickets are drawn at random with replacement from a box containing 100 tickets with different numbers, of which 25 are red, 25 black, 25 purple, and 25 blue. What is the probability that the second ticket is red?

p(second read)= 1/4

2.4. Among employees of a certain firm, 70% know C/C++, 60% know Fortran, and 50% know both languages. What portion of programmers (a) does not know Fortran? (b) does not know Fortran and does not know C/C++? (c) knows C/C++ but not Fortran? (d) knows Fortran but not C/C++? (e) If someone knows Fortran, what is the probability that he/she knows C/C++ too? (f) If someone knows C/C++, what is the probability that he/she knows Fortran too?

a) 0.4 b) 0.2 c) 0.2 d) 0.1 e) 0.833 f) 0.714

2.21. In the system in Figure 2.7, each component fails with probability 0.3 independently of other components. Compute the system's reliability.

a) 0.49 b) 0.7399 c) 0.657 d) 0.2601 e) 0.51

2.25. This is known as the Birthday Problem. (a) Consider a class with 30 students. Compute the probability that at least two of them have their birthdays on the same day. (For simplicity, ignore the leap year). (b) How many students should be in class in order to have this probability above 0.5?

a) 0.70632 b) 0.5073

2.11. A computer program is tested by 5 independent tests. If there is an error, these tests will discover it with probabilities 0.1, 0.2, 0.3, 0.4, and 0.5, respectively. Suppose that the program contains an error. What is the probability that it will be found. (a) by at least one test? (b) by at least two tests? (c) by all five tests?

a) 0.8488 b) 0.4774 c) 0.0012

3.19. A and B are two competing companies. An investor decides whether to buy (a) 100 shares of A, or (b) 100 shares of B, or (c) 50 shares of A and 50 shares of B. A profit made on 1 share of A is a random variable X with the distribution P(X = 2) = P(X = −2) = 0.5. A profit made on 1 share of B is a random variable Y with the distribution P(Y = 4) = 0.2, P(Y = −1) = 0.8. If X and Y are independent, compute the expected value and variance of the total profit for strategies (a), (b), and (c).

a) 40000 b) 40000 c) 0 Var(50x+50y)= 20000

2.22. Three highways connect city A with city B. Two highways connect city B with city C. During a rush hour, each highway is blocked by a traffic accident with probability 0.2, independently of other highways. (a) Compute the probability that there is at least one open route from A to C. (b) How will a new highway, also blocked with probability 0.2 independently of other highways, change the probability in (a) if it is built

a) p(BC)=0.95232 b) NA

Consider a fair coin which when tossed results in either heads (H) or tails (T). The coin is tossed two times. (a) Write the sample space (Order matters here. So, HT and TH are not the same outcome). (b) List all possible events and compute the probability of each event, assuming that the probability of each possible outcome is equal (Keep in mind that there should be many more events than outcomes and not all events will have the same probability).

a) sample space={HH, HT, TH, TT} b) p(HH)=1/4 p(HT)=1/4 p(TH)=1/4 p(TT)=1/4 p(one H)= 3/4 p(one T)= 3/4 p(HH&TT)= 2/4

3.2. Every day, the number of network blackouts has a distribution (probability mass function) A small internet trading company estimates that each network blackout results in a $500 loss. Compute expectation and variance of this company's daily loss due to blackouts.

loss daily due to black out $200 var(I)= 110000

In one year, three awards (research, teaching, and service) will be given to a class of 25 graduate students in a math department. If each student can receive at most one award, how many possible selections are there?

p(25, 3)= 13,800

#6. A diagnostic test for a certain disease has 95% sensitivity (a positive test given the person has the disease) and 95% specificity (a negative test given the person does not have the disease). Only 1% of the population has the disease in question. Given the diagnostic test reports that a person chosen at random from the population tests positive, what is the probability that the person does, in fact, have the disease? Are you surprised by the size of the answer? Do you consider this diagnostic test reliable?

p(A|B)= 0.1610

2.17. A computer assembling company receives 24% of parts from supplier X, 36% of parts from supplier Y, and the remaining 40% of parts from supplier Z. Five percent of parts supplied by X, ten percent of parts supplied by Y, and six percent of parts supplied by Z are defective. If an assembled computer has a defective part in it, what is the probability that this part was received from supplier Z?

p(D)=0.072 p(z found D)= 1/3

2.19. At a plant, 20% of all the produced parts are subject to a special electronic inspection. It is known that any produced part which was inspected electronically has no defects with probability 0.95. For a part that was not inspected electronically this probability is only 0.7. A customer receives a part and find defects in it. What is the probability that this part went through an electronic inspection?

p(D|PN)= 0.3 P(D|PI)= 0.04

#7. In a particular community, 70% of the voters are Democrats, and 30% are Republicans. Sixty percent of the Democratic voters and 40% of Republican voters favor the incumbent. What is the probability that a randomly selected voter from this community favors the incumbent?

p(I)=0.54

2.20. All athletes at the Olympic games are tested for performance-enhancing steroid drug use. The imperfect test gives positive results (indicating drug use) for 90% of all steroid-users but also (and incorrectly) for 2% of those who do not use steroids. Suppose that 5% of all registered athletes use steroids. If an athlete is tested negative, what is the probability that he/she uses steroids?

p(N|S)=5/936

3.6. A computer program contains one error. In order to find the error, we split the program into 6 blocks and test two of them, selected at random. Let X be the number of errors in these blocks. Compute E(X).

p(error)= 1/6

2.8. A shuttle's launch depends on three key devices that may fail independently of each other with probabilities 0.01, 0.02, and 0.02, respectively. If any of the key devices fails, the launch will be postponed. Compute the probability for the shuttle to be launched on time, according to its schedule.

p(on time)=0.95