Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)Problem Set 1Due: September 10, 2008Note: Problem set solutions are due at the start of the lecture on the date noted above, andmust be turned in within 15 minutes of the start of the lecture, i.e. by 12:15 pm, at thelatest. Late solutions will not be accepted beyond this time. There will be no exceptions tothis rule (other than in the event of an emergency, in wh ich case a Dean’s letter will be required).Therefore, it is better to hand in whatever you have done by that time!Reading: Bertsekas and Tsitsiklis, Introduction to Probability, Sections 1.1 and 1.2.1. Express each of the following events in terms of the events A, B and C as well as the operationsof complementation, union and intersection:(a) at least one of the events A, B, C occurs;(b) at most one of the events A, B, C occurs;(c) none of the events A, B, C occurs;(d) all three events A, B, C occur;(e) exactly one of the events A, B, C occurs;(f) events A and B occur, but not C;(g) either event A occurs or, if not, then B also does not occur.In each case d raw the correspond ing Venn diagrams.2. Let A and B be two events. Use the axioms of probability to pr ove the following:(a) P (A ∩ B) ≥ P (A) + P(B) − 1(b) Show that the probability that one and only one of the events A or B occurs is P (A) +P (B) − 2 · P (A ∩ B).Note: You may want to argue in terms of Venn d iagrams, but you should also provide acomplete proof, that is a step-by-step derivation, where each step appeals to an axiom or alogical ru le.3. Find P (A ∪ (Bc∪ Cc)c) in each of the following cases:(a) A, B, C are mutually exclusive events and P (A) = 3/7.(b) P (A) = 1/2, P (B ∩ C) = 1/3, P (A ∩ C) = 0.(c) P (Ac∩ (Bc∪ Cc)) = 0.65.4. Anne and Bob each have a d eck of playing cards. Each flips over a randomly selected card.Assume that all pairs of cards are equally likely to be drawn. Determine the following prob-abilities:(a) the probability that at least one card is an ace,(b) the probability that the two cards are of the same suit,Page 1 of 2Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2008)(c) the probability that neither card is an ace,(d) the probability that neither card is a diamond or club.5. Alice and Bob each choose at random a number between zero and one according to theuniform probability law. Consider the following events:A = {The magnitude of the difference of the two numbers is greater than 1/3.}B = {At least one of the numbers is greater than 1/4.}C = {The sum of the two numbers is 1.}D = {Alice’s number is greater than 1/4.}Find the following probabilities: P (A), P (B), P (A ∩ B), P (C), P (D), P (A ∩ D).6. After an ins pirational 6.041 lecture you are walking behind building 34 when you spot theMIT Beaver running a game of chance. Being astute in the ways of probability you walkover to investigate. Mr. Beaver claims to possess magical 4-sided dice1. He tells you theprobability of any particular outcome is proportional to the product of the number on eachdie, and all outcomes that result in a particular product are equally likely. He proposes thefollowing game. He rolls the dice once, if the sum of the dice is less th an or equal to 4 youwin else you lose. Wh at’s the probability you win?7. These five stocks closed at the following prices per share at the end of August 29:IBM $121.73Microsoft $27.29Apple $169.53Intel AMS $23.97Google $463.29Give us your opinion (or best guess) as to whether the share price of each will rise or fallby the end of November. Put a plus next to the company name if you believe the closingprice on November 28 will be greater than or equal to the closing price above for the end ofAugust. Put a minus next to the company name if you believe the closing price will be lessthan the closing price above. Feel free to guess. We will use these answers in the statisticssection of the course in December (and award a pack of Coke Classic to the best predictors.)G1†. Let A, B, C, A1, . . . , Anbe some events. Show the following identities. A mathematical deriva-tion is required, but you can use Venn diagrams to guide your thinking.(a) P (A ∪ B ∪ C) = P (A) + P (B) + P (C) − P (A ∩ B) − P (B ∩ C) − P (A ∩ C)+P (A ∩ B ∩ C),(b) P (∪nk=1Ak) = P (A1) + P (Ac1∩ A2) + P (Ac1∩ Ac2∩ A3) + · · · + P (Ac1∩ · · · ∩ Acn−1∩ An).1Dice or a pair of dice is understood to be to 2 individual die. Each die has 4 sides, each side labeled 1 thru 4†Required for 6.431; optional for 6.041 Page 2 of
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