CS 70 SPRING 2008 — DISCUSSION #8LUQMAN HODGKINSON, AARON KLEINMAN, MIN XU1. Probability Space: The Final FrontierExercise 1. Describe fully the probability space associated with the following random experiments, i.e,specify exactly what the sample space is and the probability associated with each sample point.(1) Aaron flips two fair coins one after another(2) Aaron flips two fair identical coins at the same time.(3) Prof. Wagner chooses three students in CS70 uniformly at random to bes tow A+.(4) Min flips a fair coin repeatedly until he sees a head.Exercise 2. Let the sample space Ω = {0, 1, 2, 3}, and let the probability of each sample point be uniform.What is the probability of the events A = {1, 2}, B = {2, 3}, C = {1, 3}? Are events A and B independent?What is P r[A|B ∪ C]?Exercise 3. Consider the case where Prof. Wagner is choosing 3 students from a class of 130 uniformlyat random and without replacement. Describe the event that you will be chosen, i.e. specify what samplepoints this event contains. What is the probability of the event that you will be chosen?Answer:2. Luqman’s Alarm and Conditional ProbabilityLuqman lives in a particularly dangerous part of Berkeley (for the sake of this problem) and at any givennight, he has a 25% probability of getting burglarized. In desperation, Luqman decided to buy a home alarmsystem.Exercise 4. Luqman bought his first alarm from a street vendor on Telegraph avenue. He checked thecircuitry of the device and discovered that it was programmed to always have 1/3 probability of going offevery night independent of whether there is burglary in progress or not. Use this information to fill out thefollowing chart: (Let A be the event that alarm is going off and B be the event that there is a burglar inthe house, and ¬A represent the complement of A, i.e. alarm is not going off)Sample Point ProbabilityA ∩ BA ∩ ¬B¬A ∩ B¬A ∩ ¬BExercise 5. After purchasing his second alarm from Craigslist, Luqman stayed up for 72 nights straight,observed his alarm, and gathered the following data:Date: April 3, 2008.1What happened? Happened for how many nights?Alarm sounded and found burglar 4Alarm sounded and no burglar 20Alarm didn’t sound and found burglar 14Alarm didn’t sound and no burglar 34Use these data to approximate the probabilities of the following joint distribution:Sample Point ProbabilityA ∩ BA ∩ ¬B¬A ∩ B¬A ∩ ¬BExercise 6. Assuming that the probability Luqman derived are accurate, use the information to calculatethe Pr[A] and P r[A|B]. Note that this example shows that knowing P r[A] and P r[B] is not enough tospecify the joint distribution. Is Luqman’s second alarm a wise purchase?Exercise 7. Take a break from Luqman’s adventure and prove that for any two events A, B of any probabilityspace, P r[A|B] + P r[¬A|B] = 1. Interprete this result intuitively. Prove also that X, Y are disjoint events,then P r[X ∪ Y |B] = P r[X|B] + P r[Y |B]. Notice that setting P rB[ω] = P r[ω|B], where ω ∈ Ω, defines anew probability space on the same sample space!Exercise 8. Continuing on from Luqman’s catastrophic second alarm. Luqman thought to himself, ”Well,since my alarm is more likely to be SILENT when there is a burglar than on average, then it should alsobe more likely to be silent when there is burglar than when there is not.” Prove that Luqman’s conjectureis correct by showing that for any probability space and any events A, B, P r[A|B] > P r[A] implies thatP r[A|¬B] < P r[A]. What happens when P r[A|B] = Pr[A|¬B]? Again, be sure to justify these resultsintuitively to yourself and not just memorize them as esoteric results of some r andom algebraic manipulations.Answer:Exercise 9. Rather than buying a new alarm system, Luqman contemplated, ”because my Craiglist alarmis more likely to be silent when there is a burglar than when there is not, I can just use my alarm in therevers e role and call the police whenever the alarm doesn’t sound!” Use conditional probability to showLuqman that this is not a good idea. (hint: calculate P r[B|¬A])Exercise 10. Supposing that the following statistics is correct, use the same reasoning from the previousexercise to explain why it is that introverted people are much more likely to be a salesman than a librarian:Type NumberExtrovert and Salesman 20 millionIntrovert and Salesman 2 millionExtrovert and Librarian 100Introvert and Librarian 50000Exercise 11. Finally, Luqman went to Radioshack and saw an alarm system that is guaranteed to have99% chance of catching a burglar, i.e. P r[A|B] = 99/100. Should Luqman buy the alarm based on just thatinformation?Exercise 12. Take another break from Luqman’s misadventure and prove that for any two events A, B ofany probability space, if event A’s occurrence makes event B more likely, i.e. P r[B|A] > P r[B], then eventB’s occurrence also makes event A more likely, i.e. P r[A|B] > P r[A].2Exercise 13. After more inquiring at Radioshack, Luqman discovered that P r[A|¬B] = 1/3 and decidedto buy the system. After installing the alarm, Luqman suddenly heard the alarm going off one night. Whatis the probability that he will catch a burglar?Answer:Exercise 14. Shortly after Luqman purchased the Radioshack alarm, the Berkeley city police increasedthe patrol around Luqman’s area and the probability that the thief will visit Luqman at any night is nowreduced to 1%. Suppose P r[A|B] stays constant because of the circuitry inside the alarm, will P r[A] increaseor decrease? What about P r[B|A]?Exercise 15. After a year, the Berkeley police department made some secret patrol changes. Now, Luqmanobserves that on any given night, his alarm has 2/3 chance of going off. Did the Berkelely PD increaseor decrease the patrol around Luqman’s apartment? What is the new probability that Luqman will getburglarized on a
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