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EBF 472: Quantitative Analysis in Earth Sciences Problem Set #1




Question;Include work and/or an explanation to justify ALL ANSWERS. That is, dont justwrite the final answer, but make sure you show how and/or why you got it.All probabilities should be rounded to two decimal places, unless indicated otherwise.Be neat and organized. If youre not, expect an immediate 20% grade reduction.Point values for each question are in parentheses.1.(10 pt) Assume that you have a standard deck of 52 playing cards. A probabilityexperiment is drawing a card from the deck. Define Event A as drawing a red card.Define Event B as drawing a face card (that is, a jack, queen, or king).Given these definitions, describe the events below in words (that is, just asEvents A & B are described in words in the previous paragraph). Also, for each of theevents below, indicate how many cards make up that event (for example, 26 cardsmake up Event A because there are 13 diamonds and 13 hearts in a deck of cards):a.b.c.d.e.BCAB(A B)CAB(A B)C2.(10 pt) Assume events E1, E2, E3, E4 and E5 are mutually exclusive andcollectively exhaustive, and that Pr{E1}=0.1, Pr{E2}=0.1, Pr{E3}=0.2, and Pr{E4}=0.2.a.b.c.d.e.What is Pr{E5}? How do you know?Draw a space-filling Venn diagram on a sample space S representing these events.Find Pr {E1 U E2 U E3}. Explain your answer in a sentence.Find Pr { (E2 U E4)C}. Explain your answer in a sentence.Find Pr { E 1 E 4 }. Explain your answer in a sentence.3.(15 pt) In class, I showed a geometric way to derive the Additive Law of Probability forthree events. Here, prove mathematically that for events E1, E2 and E3:Pr{E1 E 2 E3 } = Pr{E1 } + Pr{E 2 } + Pr{E3 } Pr{E1 E3 } Pr{E1 E 2 } Pr{E 2 E3 } + Pr{E1 E 2 E3 }To do this, you will need to use the Additive Law of Probability for two events, but thinkof one of the events in the Additive Law as a compound event (the union of twoevents). Also, you will need to use the following two properties of unions andintersections to complete the problem (consider them hints):Hint #1Hint #2A (B C) = (A B) (A C)(A B) (B C) = A B C4.C(15 pt) In class, I showed that Pr{ (E1 E2) C } = Pr{E1C E2 } using both a VennDiagram approach and a specific example involving two events: Event E1 =Temperature > 70oF on a given day, and Event E2 = It rains on a given day. Usingboth of these methodologies and the events as defined above, demonstrate theCCCvalidity of the second of DeMorgans Laws: Pr{ (E1 E2) } = Pr{E1 E2 }5.(10 pt) One question that meteorologists get a lot goes something like this: Lastwinter was snowy, does that mean this winter will be snowy too? Basically, thisquestion explores the idea of whether snowfall during a given winter depends on howmuch it snowed the previous winter. Here, you will investigate this idea using snowfalldata from State College.Consider the file StateCollegeWinterSnow.xls (posted on ANGEL), which giveswinter snowfall data for State College going back to the winter of 1896-97 (a total of115 winters). Fifty inches of snow in one winter is not all that common in State Collge,and most folks living here would consider a winter with that much snow a snowywinter. With that in mind, define Event A as fifty inches or more of snow in StateCollege in winter. Use the data to answer the following:a.Estimate Pr {A}.b.Estimate Pr {A | previous winter had 50 inches of snow or more }.c.Estimate Pr {A | previous winter had less than 50 inches of snow }.d.Do your estimates of conditional probability suggest theres a positive statisticaldependence from one winter to the next in the snowfall data? Justify your answer.6.(5 pt) The International Space Station (ISS) orbits 350 km up, but the ISS is so largethat it can be seen from Earth with the naked eye, if you know when and where tolook. For this problem, assume that on a clear night, a properly trained observer has a90% chance of seeing the ISS when it passes overhead.Assume that the ISS will be passing overhead tonight in State College, andskies will be clear. Ten (properly trained) members of the Penn State Astronomy Clubposition themselves at ten different locations in State College, each independentlyattempting to see the ISS.a.What is the probability that all ten members of the club will see the ISS tonight?b.What is the probability that none of the members of the club will see the ISS tonight?For this part of the problem, keep 5 decimal places accuracy in your answer.7.(Adapted from Ch.2 of Wilks (2005)): (15 pt) There is some evidence that droppingcertain chemicals into a thunderstorm (a process known as cloud seeding) mayreduce the amount of damaging hail that a thunderstorm produces.Assume that the effect of cloud seeding on the suppression of damaging hail isbeing studied by randomly seeding or not seeding equal numbers of thunderstorms(that is, half the thunderstorms are seeded and half are not). Suppose that theprobability of getting damaging hail from a seeded thunderstorm is 0.15, while theprobability of getting damaging hail from an unseeded thunderstorm is 0.45. If one ofthe thunderstorms in the study has just produced damaging hail, what is the probabilitythat it was seeded? In your answer, keep three decimal places accuracy.8.(10 pt) Consider the file StateCollegeAltoonaTemp.xls, posted on ANGEL. This tablegives the maximum temperature (Tmax) in oF for each day from December 1, 2011 toJanuary 10, 2012, for State College and Altoona (which is about 35 miles southwest ofState College).In both locations, a temperature of 40oF in winter would be considered a warmday by local standards. With that in mind, define Event A as a day on which StateCollege Tmax>40oF, and Event B as a day on which Altoona Tmax>40oF. Using thedata in this table:a.In words, what does Pr{ A B } mean? What about Pr{A|B}?b.Estimate Pr {A}, Pr {B}, and Pr{ A B } from the data.c.Using an appropriate formula and the results from part (b), calculate Pr {A|B}.d.Explain how you would compute Pr {A|B} directly from the data.e.Use your results to decide whether Events A and B are independent, and explain howyou came to your conclusion.9.(10 pt) This question explores the notion of persistence as a problem in conditionalprobability. We will use the same temperature dataset used in Problem #8 andinvestigate whether the maximum temperature one day is related to the maximumtemperature on the next day.a.From the data, estimate the probability that the State College Tmax>40oF, given thatthe previous days State College Tmax>40oF.b.From the data, estimate the probability that the State College Tmax>40oF, given thatthe previous days State College Tmax


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