EE307ProbabilityandRandomProcesses:ProblemSet1

1. IfA={2x5}andB={3x6},findAUB,AB1,and(AUB)(AB)2.2. ShowthatifAB=,thenP(A)P(B).3. IfS={1,2,3,4},findthesmallestfieldthatcontainsthesets{1}and{2,3}.4. Acalloccursattimetwheretisrandompt.intheinterval(0,10).Find(a)P(6t8)andP(6t8/t>5).5. TheeventsAandBaremutuallyexclusive.Cantheybeindependent?6. Box1contains1000bulbsofwhich10%aredefective.Box2contains2000bulbsofwhich5%aredefective.

Twobulbsarepickedfromarandomlyselectedbox.(a)Findtheprobabilitythatbothbulbsaredefective.(b)Assumingthatbothbulbsaredefective,findtheprobabilitythattheycamefromBox1.

7. Wehavetwocoins;oneisfairandtheotheristwoheaded.Oneofthecoinsispickedatrandomandtossedtwice.Headsshowsupbothtimes.Findtheprobabilitythatthecoinpickedisfair.

8. Aslandlord,youareobligedtoprovidelaundryfacilitytoyourtenants.Yourchoicesaretolease1)twoinexpensivemachinesthatbreakdownfrequently(unreliable)atRs50/montheach,or2)asingle,morereliablemachineatRs100/month.Ifunreliableisoutofcommission40%ofthetimeandreliableisoutofcommission20%ofthetime,(a)Fromthetenantspointofviewwhichisthebetteralternative?(b)fromthelandlordsPoVwhichisthebetteralternative.(Incompleteinfo;youmayneedtomakesomepracticalassumptions).

9. Arandomnumbergeneratorproducesintegersfrom19(inclusive).Alloutcomesareequallylikelyandeachintegerisgeneratedindependentlyofanypreviousinteger.Let=N1+N2denotethesumoftwoconsecutivelygeneratedintegers.(a)Giventhatisodd,whatistheconditionalprobabilitythat=7?(b)Giventhat>10,whatistheconditionalprobabilitythatatleastoneoftheintegersis>7?(c)GiventhatN1>8,whatistheconditionalprobabilitythatwillbeodd?

10. Wargamestrategistssolveproblemsofthefollowingtype:Thereare6incomingballisticmissiles(BM)againstwhicharefired12antimissilemissiles(AMM).TheAMMsarefiredsothattwoofthemaredirectedagainsteachBM.Thesingleshotkillprobability(SSKP,theprobabilitythatanAMMdestroysaBM)ofanAMMis0.8.AssumethattheAMMsdontinterferewitheachotherandthatanAMMcan,atmost,destroyonlytheBMagainstwhichitisfired.Computetheprobabilitythat(a)allBMsaredestroyed,(b)atleastoneBMgetsthroughtodestroythetarget,(c)exactlyoneBMgetsthrough.

11. Asmuggler,tryingtopasshimselfoffasaglassbeadimporter,attemptstosmugglediamondsbymixingdiamondbeadsamongglassbeadsintheproportionofonediamondper1000glassbeads.Abusycustomsinspectorexaminesasampleof100beads.Whatistheprobabilitythatthesmugglerwillbecaught?

12. Acountryisatcivilwar,anditisnotuncommontofindabombonboardpassengeraircraft.Theprobabilitythatabombwillbeonboardonanygivenflightis102.Anervouspassengeralwaysflieswithanunarmedbombinhissuitcase,reasoningthattheprobabilitythattherewillbe2bombsonboardis104.Bythismaneuver,thenervouspassengerbelievesthathehasgreatlyreducedtheairplaneschancesofbeingblownup.Whatdoyouthink?

1ABisusedtodenoteAintersectionB2(AB)denotescomplementofAintersectionB.