Probability II. Overview. A Closer Look at Events The Probability of an Event. Dr Tom Ilvento Department of Food and Resource Economics

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1 Oveview Pobability II D Tom Ilveto Depatmet of Food ad Resouce Ecoomics We will cotiue ou jouey though pobability This ivolves Moe tems!!! Defiig Evets Ways to lay out the sample space I will also give you some basic ules of coutig - the lottey example Fially, some eal examples of pobability that do t ivolve cois o dice! 2 A Close Look at Evets The Pobability of a Evet A evet is a specific collectio of sample poits Fo example, o a oll of the dice Sample poits ae [1, 2, 3, 4, 5, 6] Evet A: [Eve umbes] cotais the sample poits 2, 4, 6 Evet B: [Odd umbes] cotais the sample poits 1, 3, 5 Evets ae subsets, which ca ovelap o be mutually exclusive Evet A: [Eve umbes] cotais the sample poits 2, 4, 6 Evet C: [Numbes geate tha 2] cotais the sample poits 3, 4, 5, ad 6 Evets A ad C shae the commo sample poits 4 ad 6 3 The pobability of a Evet A, deoted as P(A), is calculated by Establishig all the sample poits Establishig the pobabilities fo each sample poit summig the pobabilities of the sample poits that petai to Evet A 4

2 Basic Steps to obtai a Pobability of a Evet 1. Defie the expeimet descibe pocess of makig a obsevatio 2. List the Sample Poits 3. Assig pobabilities to the Sample Poits 4. Detemie the collectio of Sample Poits cotaied i the Evet of iteest 5. Sum the Sample Poits pobabilities to get the Evet pobability Rollig a sigle die Outcomes S: [1, 2, 3, 4, 5, 6] Pobabilities fo the outcomes Outcome 1 1/6 =.167 Outcome 2 1/6 =.167 Outcome 3 1/6 =.167 Outcome 4 1/6 =.167 Outcome 5 1/6 =.167 Outcome 6 1/6 = Rollig a sigle die Repesetig the sample space via a tee o diagam Evet: ollig a 2 Sample Poits 2 P(ollig a 2) = 1/6 =.167 Evet: ollig 3 o highe Sample Poits 3, 4, 5, 6 P(ollig 3+) = 1/6+ 1/6 + 1/6 +1/6 = 2/3 P(ollig 3+) =.667 Evet: ollig a eve umbe Sample Poits 2, 4, 6 P (ollig eve) = 1/6 +1/6 +1/6 =.5 7 We ca epeset the Sample Space ad Evets by Listig of the Set Ve Diagam Cotigecy Table Decisio Tee Diagam 8

3 Sample Poit Ve Diagam Expeimet: Flip a coi twice, ote the face each time Sample Poits [HH HT TH TT] Evet D: At Least Oe Tail EVENT D: [HT TH TT] HEAD HH HT TH TAIL TT Evet at least oe tail 9 Cotigecy Table Expeimet: Flip a coi twice, ote the face each time 1st Coi 2d Coi Head Tail Total Head HH HT HH HT Tail TH TT TH TT Total HH TH HT TT 10 Decisio Tee Diagam A Pobability Poblem Expeimet: Flip a coi twice, ote the face each time Secod Flip H HH Fist Flip H T HT H TH T T TT Outcome I have a ja cotaiig five mables, 2 of which ae blue ad 3 of which ae ed. I adomly daw two mables What is the pobability of dawig 2 blue mables? 1. Defie the expeimet 2. List o daw out the sample poits 3. Assig pobabilities to the sample poits 4. Detemie the collectio of sample poits cotaied i a evet of iteest S = {HH, HT, TH, TT} Sum the sample poits pobabilities to get the evet pobability 12

4 Mable poblem solutio Daw 2 mables whee 2 ae Blue ad 3 ae Red Mable poblem solutio Daw 2 mables whee 2 ae Blue ad 3 ae Red Sample Poits B fo blue B1 B2 B1 R1 B1 R2 B1 R3 B2 R1 B2 R2 B2 R3 R1 R2 R1 R3 R2 R3 R fo Red What is the pobability of dawig 2 blue mables? I this poblem the ode of the mables is ot impotat to us So thee is oly oe combiatio of dawig two blue mables Uless othewise kow, each outcome has a equal pobability Thus each has a chace of beig daw The Pobability of dawig two Blue Mables is = Sample Poits B fo blue B1 B2 B1 R1 B1 R2 B1 R3 B2 R1 B2 R2 B2 R3 R1 R2 R1 R3 R2 R3 R fo Red What is the pobability that a blue ad a ed mable ae daw? Thee ae six sample poits that satisfy this Evet The Pobability of dawig a blue ad a ed mable is 6/10 = Mable poblem solutio Daw 2 mables whee 2 ae Blue ad 3 ae Red Sample Poits B fo blue B1 B2 B1 R1 B1 R2 B1 R3 B2 R1 B2 R2 B2 R3 R1 R2 R1 R3 R2 R3 R fo Red What is the pobability that two ed mables ae daw? Thee ae thee sample poits that satisfy this Evet The Pobability of dawig two ed mables is 3/10 = Let s e-expess these sample poits as combiatios of blue ad ed mables Sample Poits Two Blue Blue & Red Two Red Pobability 6/10 3/10 The sample space ca be expessed as a set of mutually exclusive evets 16

5 Coutig Stategies A key step fo may of these pobability poblems is to idetify the sample space. Thee ae may stategies to cout the combiatios of thigs, icludig: Fudametal coutig ule sequece of two evets i which the fist evet occus m ways, ad the secod evet ways, the evets togethe occu m* ways Factoial aagig thigs all possible ways. Factoials ae oted as! 3! = 3*2*1 = 6 Pemutatios the umbe of sequeces of items selected fom available items (without eplacemet), ad the ode is impotat, give as P =!/(-)! Combiatios the umbe of combiatios of items fom diffeet items (without eplacemet), ad the ode does t matte C =!/!(-)! If you daw 2 cads fom a stadad deck of 52 cads without eplacemet, thee ae 52 ways to daw the fist ad 51 ways to daw the secod, equal to 52*51 o 2652 ways to daw the two cads How may hadshakes i a oom of 6 people? it is 6! = 6*5*4*3*2*1 = 720 How may ways ca 9 hoses fiish a ace i tems of the top thee ode - fist, secod ad thid? This is 9 objects take 3 at a time: 9P3 = 9!/(9-3)! = Combiatio Rule C To fid the umbe of samples of thigs take at a time The objects ae distict Oce a object is used it ca t be epeated ode is ot impotat Whee! is Example: 5!! Note: 0! = 1 ad 1! = 1 Fo ou Mable Poblem we have 5 mables ad choose two =5 ad =2 ' % & $! " = = #!(! )!! = (! 1)(! 2)...(1) C 5 = 5(5! 1)(5! 2)(5! 3)(5! 4) = 120 5C 2 = 120 2(3!) = 120 2(6) = =10 5! 2!(5! 2)! 18 Pobabilities This is a lottey game whee you must coectly guess 5 white ball umbes out of 59 balls 1 ed ball umbe out of 39 balls The odds ae give as: 1 out of 195,249,054 How did they get that umbe??? To coectly guess the white ball umbes, it is 5 out of 59 59C5 = 59!/5!(59-5)! = 5,006,386 To wi you must also coectly guess the Red Ball 39C1 = 39!/1!(39-1)! = 39 Both evets must happe at the same time, ad they ae idepedet of each othe, so we multiply the two pobabilities to get the fial aswe: P(wiig) = 5,006,386*39 = 195,249, Cosume Electoics Study Suvey of 1,000 households Obtaied fom a adom pocess Households ae thought to be idepedet of each othe Two questios ae asweed: Whethe the household had plaed to puchase a Big Scee TV (BSTV) Whethe the household actually puchased a BSTV ove the yea 20

6 Results fo Cosume Electoics Study Actually Puchased a BSTV Isect Pheomoe Study Plaed to Puchase BSTV Yes No Total Yes No Total ,000 Let A = Pobability of Pla to Puchase a BSTV P(A) = 250/1000 =.25 Let B = Pobability of actually puchased a BSTV P(B) = 300/1000 =.30 The pobabilities based o the umbes withi the table ae based o combiatios of plaig ad actually puchasig a BSTV The aswe moe iteestig questios: what pecet of those tha plaed to buy a BSTV actually puchased oe? 200/250 =.80 o 80% We will evisit these poblems late 21 Etomologists ae ofte iteested i studyig the effect of chemical attactats (pheomoes) o isects. Oe commo techique is to elease seveal isects equidistat fom the pheomoe beig studied ad a cotol substace. If the pheomoe has a effect, moe isects will tavel towad it tha towad the cotol. Othewise, the isects ae equally likely to tavel i eithe diectio. Suppose five isects ae eleased, deoted as isect A, B, C, D, E How may outcomes (simple evets) ae possible, if the focus is o how may isects tavel towad the pheomoe? 22 Isect Pheomoe Study Isect Pheomoe Study Thik of the Sample Space as the umbe of isects tavelig towad the pheomoe. We ca cout the diffeet outcomes Noe Oe: A B C D E Two: AB AC AD AE BC BD BE CD CE DE Thee: ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE Fou: ABCD ABCE ABDE ACDE BCDE Notice that each of these outcomes ae based o the Combiatio Rule. How may combiatios of 3 out of 5 isects? 5C3 = 5!/3!(5-3)! = Five: ABCDE 0 Noe All togethe thee ae 32 diffeet outcomes that ca happe Suppose the pheomoe ude study had o effect ad, theefoe, it is equally likely that a isect will move towad the pheomoe (o ot). Fid the pobability that both isects A ad E tavel towad the pheomoe This Evet is based o the followig poits AE ABE ACE ADE ABCE ABDE ACDE ABCDE This is 8 out of the 32, o 8/32 =.25 Just by chace, we would expect that 25% of the time that both isects A ad E will tavel to the pheomoe We could coduct the expeimet ad compae ou obsevatios to the chace esult to see if thee is a diffeece A diffeece meas thee is somethig moe tha chace wokig hee Because this is a sample, we ca t be sue about small diffeeces 24

7 Suvey of Busiesses Suvey of Busiesses A eseache wated to fid the pimay easo fo a compay to egage i divesity taiig She suveyed busiesses via a adom sample to detemie the pimay easo fo divesity taiig, offeig five mutually exclusive, ad exhaustive optios Listed ae the pecetages fo each - they ca be thought of as pobabilities Reaso Pecetage Comply with pesoel policies (CPP) 7% Icease poductivity (IP) 47% Stay competitive (SC) 38% Social esposibility (SR) 4% Othe (O) 4% 25 What is the pobability that the pimay easo fo divesity taiig is busiess elated; i.e., elated to competitio o poductivity? Let Evet B = divesity is busiess elated P(B) = P(IP+SC) = =.85 What is pobability that social esposibility is ot a pimay easo fo divesity taiig? Let Evet C = Not SR P(Not SR) = P(CPP+IP+SC+O) = =.96 Reaso Pecetage Comply with pesoel policies (CPP) 7% Icease poductivity (IP) 47% Stay competitive (SC) 38% Social esposibility (SR) 4% Othe (O) 4% Fo the last aswe We could have thought of this as the complemet of SR Deoted as SR C SR C = 1 P(SR) SR C = =.96 The complemet of a evet A is the evet that A does ot occu that is all sample poits ot i Evet A Deoted as A C o as A 26 Summay We exteded ou pobability vocabulay We leaed how to epeset the sample space by tables, Ve Diagams o decisio tees Ad we itoduced some coutig ules Fo some poblems, just kowig the total of the diffeet outcomes ca be a majo task The oe coutig ule I will equie you to kow is the Combiatio Rule ' % & Now we ae eady to tackle some of the basic ules of pobability that elate to uios, itesectios ad coditioal pobability $! " = = C #!(! )! 27