Homework Set # Part I: COMBINATORICS (follows Lecture ). The Supreme Court (9 members) meet, and all the justices shake hands with each other. How many handshakes are there? 2. A country has license plates made up of 3 letters, followed by 3 digits. a. How many plates are possible? b. How many plates are possible if letters cannot be repeated? c. How many plates are possible if the 3-digit number must be an odd number? (e.g., ACQ 245) 3. Five red books and four green books (all with different titles) are to be arranged on a shelf. a. How many arrangements are possible altogether? b. How many arrangements are possible if the red books must be all together? 4. a. In the sketch below, how many minimal length paths are there from A to B? b. How many are there if you can't use the link between C and D? A C D 5. How many ways can you choose 5 hearts from a 52-card deck? Note: There are 3 hearts in a 52-card deck. 6. How many 5-card poker hands (from a 52-card deck) contain the Queen of Spades? (Note: There is only one Queen of Spades in a deck!) 7. How many possible ways are there to choose a committee of 5 professors from a department of 2 professors? 8. If a man takes four shirts, three pairs of slacks, and two pairs of shoes on a trip out of town, how many different outfits could he possibly wear? 9. In how many ways is it possible to seat 8 people around a circular table? B
2 Part II: BASIC PROBABILITY. (Lecture ) If two dice are tossed, find the probability of getting a 7 or. 2. (Lecture ) Find the probability of getting a Queen of Spades in a 5-card poker hand. 3. (Lecture ) Four women check their coats at a restaurant. The coats are later returned at random. Find the probability that each woman gets her own coat back. 4. (Lecture ) A certain computer system uses passwords consisting of a single letter or a letter followed by as many as 3 symbols which may be letters or digits: e.g., Z, ZZ6, JIMB, R2DT, etc. Assume that you can only use capital letters, and that all passwords are equally likely to be chosen. What is the probability that two people, John and Mary, choose the same password? 5. (Lecture 2) A machine has 7 identical independent components. The probability that a component fails is.2. In order for the machine to operate, at least 6 of its 7 components must work. Find the probability that the machine fails. 6. (Lecture 2 Puzzle Problem) In a hospital nursery, which has, on the average, as many boys as girls, the girls have not been counted; but the number of boys is 2. A nurse brings in a new baby, then randomly selects a baby from the total. The baby selected is found to be a boy. What is the probability that the added baby was a girl? 7. (Lecture 2) Suppose that.5% of the population has disease D. There is a test to detect the disease. A positive test result is supposed to mean that you have the disease, but the test is not perfect. For people with D, the test misses the diagnosis 2% of the time; that is, it reports a false negative. And for people without D, the test incorrectly tells 3% of them that they have D; that is, it reports a false positive. Suppose that you take the test and it comes back positive. What is the probability that you actually have D? 8. (Peebles; Lecture 2) A missile launches accidentally if 2 relays, A and B, fail simultaneously. The probabilities of A and B failing are.0 and.03 respectively; but B is twice as likely (prob..06) to fail if A has failed. a. What is the probability of an accidental launch? b. What is the probability the A will fail if B has failed? c. Are the events "A fails" and "B fails" independent events?
3 9. (Ross; Lecture ) The probability of winning a certain dice game on a single toss is.2. Person A starts, and if he fails he passes the dice to Person B, who tries to win on his toss. They keep tossing the dice back and forth, taking turns until someone wins. What is Person A's chance of winning? 0. (Peebles; Lecture 2) Spacecraft are expected to land in a prescribed recovery zone 80% of the time. Over a period of time, 6 spacecraft land. a. Find the probability that none land in the prescribed zone. b. Find the probability that at least one lands in the prescribed zone. c. The program is called a success if the probability is at least.9 that three or more of the six spacecraft land in the prescribed zone. Is the program successful?. (Haddad) Consider sending packets through a switch with two packetswitched channels (Channel A and Channel B). Suppose that 40% of the packets are routed via Channel A, and suppose: % of the packets are lost on Channel A; and 0.5% of the packets are lost on Channel B. a. If one packet is sent, find the probability that it is lost. b. If a packet is lost, what is the probability that it was sent via Channel A? c. If three packets are sent, what is the probability that all three packets are lost? 2. (Niven; Lecture ) If the 26 letters of the alphabet are written down in random order, what is the probability that x and y are adjacent? 3. (Lecture ) In 5-card poker, what is the probability of getting a flush? Hint: a flush is 5 cards, all of the same suit, with no requirement on the rank of the cards; e.g., 4, 2, K, 6, 9. 4. (Ross; Lecture 2) English and American spellings are "rigour" and "rigor", respectively. A man in a Parisian hotel writes this word, and a letter taken at random is found to be a vowel. If 40% of the English-speaking men at the hotel are English and 60% are American, what is the probability that the writer is an Englishman? 5. (Ross; Lecture 2) Suppose that 5% of men and.25% of women are colorblind. A color-blind person is chosen at random. What is the probability that this person is male?
4 6a. If five dice are tossed, what is the probability that the five numbers (showing on the faces of the dice) will all be different? 6b. Write a single paragraph explaining how you could use MATLAB to verify your answer. (Explain what you will have MATLAB do.) 6c. Then write a MATLAB m-file to verify your answer to (a). 7. (Ross; Lecture 2) In answering a question on a multiple choice test, say that a student knows the answer with probability 3/4. If he doesn't know, he guesses, and he gets the right answer with probability /4 if there are four possible choices. What is the probability that the student actually knew the answer, given that he answered a certain question correctly? That is, what is the probability that it wasn't just a lucky guess? 8. Suppose that we are connecting two points A and B with three sections of fiber-optic cable in series (as shown below) for the purpose of communication. The three sections were chosen at random from a box containing fifteen sections, five of which were defective. Unfortunately, we will be unable to communicate if one or more of the sections chosen is defective. Find the probability that we can communicate successfully across the link. A B 9. (Lecture ) Suppose that each child born to a couple is equally likely to be a boy or a girl, independently of the sex distribution of the other children in the family. For a couple with 5 children, find the probabilities of the following events. a. All children are of the same sex. b. The 3 oldest are boys, and the others are girls. c. Exactly 3 are boys. d. At least is a girl. 20. (Lecture 2) A computer communications network is shown below: B Computer # A C D Computer #2 Instead of a paragraph, you may choose to make a flowchart.
5 A, B, C, and D are nodes of the network. The four nodes each have a probability of failure of.2, and the node failures occur independently of each other. The network fails only if neither of the two possible paths is available. a. Find the probability that the network is successful; that is, find the probability that the two computers can communicate. b. If computer # is communicating successfully with computer #2, what is the probability that node B has failed? 2. (Lecture 2) In a communications channel, transmitted 's are incorrectly received as 0's with probability 0., while transmitted 0's are incorrectly received as 's with probability.2. If 60% of the transmitted symbols are 0's, find: a. the probability of receiving a 0 b. the probability that zero was actually transmitted, given that zero was received. 22a. (MATLAB) Write an m-file simulation to verify your answer for Part II, Problem. In other words, write an m-file that simulates the tossing of a pair of dice, and computes the fraction of outcomes for which the sum is 7 or. Complete the table below showing the fraction obtained, depending upon how many trials of the experiment are executed in your simulation. Number of Trials Pr(7 or ) 0 00 000 0,000 Theoretical Answer from Problem 22b. (MATLAB) Modify your program used in problem 22a to find the probability of getting a sum less than 7 by simulating 5,000 trials. 23. (MATLAB) Consider an experiment in which we first roll a die, and then toss a coin a number of times. If the number showing on the die is N, we toss the coin N times.assume that both the die and the coin are fair. Let Y be the number of heads obtained in the coin toss. Use MATLAB to write an m-file that simulates this experiment, and records Y for each trial. Write your m-file to simulate 0,000 trials. Based on your simulation, find:
6 a. Pr(no heads) = Pr(Y = 0) b. Pr(3 heads) = Pr(Y = 3) 24. System Design Problem: Problem 20 can be though of as a network with redundancy 2, since there are two parallel paths (through nodes B and C) connecting computers # and #2. How many parallel paths would be needed (i.e., what would the level of redundancy be) to obtain a probability of success of at least.62, assuming that all parallel paths still must pass through nodes A and D? 25. System Design Problem:A binary symmetric communications channel is shown below. Assume that the symbols 0 and are equally likely to be transmitted. Note that the cross-over probability for each symbol is (-p). Find the minimum acceptable value for p if the total probability of error must be no greater than.00. 0 p 0 p 26. (Lecture 2) Cooper & McGillem, problem -4.8 27. (Lecture ) Cooper & McGillem, problem -6.3 a, b, and c 28. (Lecture 2) Cooper & McGillem, problem -6.5 29. (Lecture 2) Cooer & McGillem, problem -7. 30. (Lecture 2) Cooper & McGillem, problem -7.2 3. (Lecture 2) Cooper & McGillem, problem -7.3 32. (Lecture 2) Cooper & McGillem, problem -7.5 33. Cooper & McGillem, problem -0.8a 34. Cooper & McGillem, problem -0.9 35. (Lecture ) a. Consider a quaternary alphabet with 4 equally-likely symbols: {0,, 2, 3). Find the entropy of the alphabet. b. Find the entropy of the same alphabet if the symbol probabilities are: Pr(0) = Pr() = /4; Pr(2) = /8; Pr(3) = 3/8. Also determine which of the 4 symbols carries the most information.
7 36. (Lecture ; Mix) An experiment consists of tossing two fair coins, so that the sample space is S = {HH, HT, TH, TT}. Find the entropy in this experiment. (Hint: first find the information associated with each outcome, and then find the entropy or average information for all the outcomes.)