Expected Value. Background

Transcription

1 Please note: Before I slam you wth the notaton from Chapter 9 - Secton, I want you to understand how smple Mathematcal Expectaton really s. My frst smplfcaton: I wll refer to t as Expected Value (E )from ths pont. Objectves: At the end of ths lesson, you should be able to:. Defne an expected value.. Calculate expected value based on a probablty model.. Evaluate the wsdom of playng a game usng expected value. Background The other day I was confronted wth a dose of realty. My lfe expectancy s 74 years. My wfe has a 79-year expectancy. Let s not get nto whether men have shorter lves (on average) because they are marred to women. These are averages, expectatons, expected values. I doubt that my lfe wll abruptly termnate on my 74 th brthday. Another grm dose of realty: My nsurance company thnks there s a possblty that I wll be nvolved n a serous accdent next month. They expect me to be a negatve cash flow to them when t happens. Maybe I shouldn t drve next month. Then agan, my chances of beng njured whle stayng home are even greater! That last sentence seems to lnk probablty and expected value. The stuatons above are parts of daly lfe. They balance what could happen wth what we expect to happen. I could be ht by a meteor tonght. I don t expect t to happen. Meteorte actvty on earth s relatvely unform today, wth an average of about one meteorte per year fallng every 7,700 square mles (,500 sq. km). There s a great deal of fuzzness n the varous statements above. Frst, lfe expectancy does depend on genetc factors, occupatonal factors, and certanly could be nfluenced by plan dumb luck (random factors lke your spouse s a seral kller or a bad drver)! My lkelhood of an accdent depends on my age, my vehcle s condton, my drvng habts, and whether you are talkng on your cell phone whle drvng near me. My chances of gettng wacked on the head by a meteorte ncrease dramatcally durng the Pleades shower (occurrng from early June and to md-july). They reach an hourly rate of 5 meteortes! I ve watched them. Not a sngle one ht me. Let s get nto the mathematcs of expected values. Frst let s do an exercse n calculatng somethng very famlar to you, your course grade. Let s assume the grade s made of homework, tests, quzzes and a fnal exam. Here s a typcal weghtng system found n many unversty syllab: Evaluaton Homework Average Test Average Quz Average Fnal Exam Weght 40% 5% 5% 0% Arzona State Unversty, Department of Mathematcs and Statstcs of 7

2 Now suppose a student took three tests, scorng 77, 85 and 9 out 00. We all know the test average s = 85 Now the student also took 0 quzzes. Each was worth ten ponts. We recorded them n a table: out of 00. So, 85% s the average. Quz Score We can ether add them up, as we dd n the test calculaton, or notce that we have repeated values. Let s use the former to do the calculaton. Ths s sometmes called a weghted average. (7) + 4(8) + (9) + (0) 8 So the quz average s = = 8. out There s a pont of order here. The student s average score on quzzes s 8.. However, we now have tests out of a 00 and quzzes out of 0. It would be smart to wrte both as a percent so they compare farly. The student has an 8% quz result and an 85% test average. Let s assume the student earned a 97% average on homework and a fnal exam grade of 89%. So what s the fnal course grade? Your gut nstnct mght be to add the four grades; that would gve an 88% average. But you would be wrong. We need to weght the components accordng to our syllabus standards. The calculaton follows: 40% of Homework Average + 5% of Test Average + 5% of Quz Average + 0% of Fnal Exam Also, remember we use percentages as a decmal value n multplcaton. So we get ths 0.40(97) + 0.5(85) + 0.5(8) + 0.0(89) = 90% Ths s another weghted average. Notce that by just averagng, you overvalued the quz result and undervalued the homework. The student has a 90% average, defntely an A grade. You may wonder why we dd all ths. Well, now you know how to calculate your own grade when the professor uses weghted values n the syllabus. And, all the math we dd here s what we wll do n calculatng an expected value. Let s start wth the tests. Suppose that we change our vew of the results. The student has three grades of 75, 85 and 9. Each has equal weght. Hs probablty for any of these grades s n. Hs expected value for repeated testng s hs arthmetc average f we make the reasonable assumpton that the three tests don t change much = ( 77) + ( 85) + ( 9) = 85 Now look at the quz score average. Hs probablty of scorng a partcular grade s not unform. The probablty for a 7 s four n 0. The probablty for a 9 s only two n 0. Hs expected value for repeated quzzng s hs weghted quz average. (7) + 4(8) + (9) + (0) 4 = (7) + (8) + (9) + (0) = of 7 Arzona State Unversty, Department of Mathematcs and Statstcs

3 Fnally, for the course grade tself: Expected Value The value of each score to the student depends on how t s obtaned, whether n homework, test, quz or fnal exam. Hs expected value s agan hs weghted score just as before. 0.40(97) + 0.5(85) + 0.5(8) + 0.0(89) = 90% Expected Value n Games Now let s apply ths to a probablty model. Many tmes the model reflects a game we mght want to play. The expected value s a relable way of removng percepton from consderaton. Most people wll agree to play a game wth a $00 wn f they only need to pay $. The lottery proves ths. However, expected value wll tell exactly what t says, What can you expect f you play repeatedly? Example: Suppose we are to choose from a bag wth balls numbered,,, and 4. There s one of each. Then we choose from another bag wth balls numbered, 4, and 5. The results are added together to create a pont. We ll call the two draws a turn. The sample space has 4 = outcomes possble, but there are only 6 possble ponts on any turn (4, 5, 6, 7, 8, 9). The lattce for ths game s to the rght. We can also dgest the results nto a table where t s easer to count or calculate the probabltes for each pont. That result s below. Note the symmetry. It can be useful. outcome P(outcome) We can calculate the expected value for the model just as we dd for the quzzes before: Expected Value = (4) + (5) + (6) + (7) + (8) + (9) = ,, 4, 5 7,, 4, 5 8,, 4, 5 9 4, 4, 4 4, 5 I thnk we can safely wrte a formula for expected value. I ll gve you two varatons, but they really are the same math. Ths s sometmes called the mean of the probablty model. It s a good name snce t reflects the fact that ths s an averagng process. Arzona State Unversty, Department of Mathematcs and Statstcs of 7

4 Expected Value Defned p( x ) Gven a probablty model where the probabltes for each of n numercal outcomes,, are known, the expected value E s the result of ths calculaton: n E = x p( x ) = Alternatvely, gven a stuaton where for each of n numercal outcomes,, the relatve frequency of occurrence are known, the expected value E s the result of ths calculaton: n E = x rf = What does expected value say n the play of our game? Not much really! On any sngle turn we wll get ponts of 4, 5, 6, 7, 8, or 9. But on average over many, many, many turns, we expect to see our expermental average get closer and closer to the expected value of 6.5. And then on our next turn we wll get ether 4, 5, 6, 7, 8, or 9. What a bore! Example: Let s spce ths game up a lttle. Suppose you were to wn $ for a pont of 4, 5, or 6. And for 7 or 8, they gve you $. If you get the 9 pont, you receve $5. I m sure you would play. After all, you wn every tme! Now the queston s, how much do you expect to wn? Well, ether $, $, or $5 n one turn, of course. But f you play 0, 00, or 000 tmes, what total do you expect to wn? Really the model s changng to focus on wnnngs. outcome P(outcome) x value $ $ $5 The expected value calculaton for ths s game s below. Expected Value = + + ($) + + ($) + ($5) = $.75 We cannot predct or say for certan what wll happen on any partcular turn. But, at the tenth turn, we expect to have won 0($.75) = $7.50; at the 00 th turn $75; and at the 000 th turn $,750. Not too shabby. Here s a key observaton. The person who set up the game gnored the realtes of the probabltes to set the value for each pont! The ponts of 6 and 7 should have the same dollar value. The pont of 4 s just as rare as the pont of 9. It s just that knd of nsanty that makes necessary to calculate expected value to decde whether you want to play the game. x x rf x 4 of 7 Arzona State Unversty, Department of Mathematcs and Statstcs

5 Here s where we nject more realty. All that money won has to be pad for somehow! Suppose I made you buy a tcket for each turn. Would you play f the tcket cost $.50 for each turn? Sure! You expect to wn $.75 $. 50 = $ 0.5, a quarter on each turn. What about $.00 each turn? I hope not! You expect to lose a quarter at each turn snce $.75 $.0 0 = $ 0.5. Let s talk about farness. How much should you pay to make ths game far? Well, f you pad $.75 to play, you should expect to break even, to wn or lose nothng n the long run. Now let s talk about runnng the game for a charty (lke a Las Vegas casno). The charty needs to make money. Suppose they estmate that 000 people wll play the game. They want ths game to earn $,000 dollars. How much do they charge for the game? Smple, they charge $.75 per turn. They expect to pay out $.75 each turn and keep a buck from each player. Wll they make the $,000. There s no guarantee. If the game s far n that the outcomes are randomly acheved, they expect to make the money or at least pretty close to t. Back to my nsurance company. Let s set up a hypothetcal stuaton. Way back when I was young, they assessed the probablty of my smashng my car to be about 0.5% monthly. The average cost to repar my car was about $00 durng that year. They wanted to make a proft so they needed to clear some money. Let s assume they wanted 7% proft from me. How much should they charge me (the premum) for my annual nsurance polcy? My cost to the nsurance company s calculated to the rght. We just calculate the expected value by assumng that I cost them 00 dollars 0.5% of the tme and nothng otherwse. Ths assumes the nsurance company makes nothng. Fat Chance! To fnsh the calculaton, we multply the zero-sum value, X, by.07% to get the 7% proft. My premum should be about $8.06 monthly, or $7.7 annually! 0.05( 00) ( X ) = ( 00) X = Premum=.07(6.68) = $8.06 Of course the numbers above are fancful. Whle I was a much better drver than that, on no less than three occasons I was ht by someone else n the same year, n three dfferent years, none of them from my error! Statstcally, I should probably by drvng a coffn now. However, the concept s not far fetched. The nsurance world uses averages for accdent rates at a partcular age, cost to repar a car, lkelhood a certan model of car wll become entangled wth another, etc., to calculate an expected value for each one of us. Then they tack on the proft they need to keep the stockholders happy. And, by the way, marred men and women are expected to lve longer than ther sngle counterparts! That tdbt factors nto the cost of your lfe nsurance. Maybe men don t lve as long as women as much because they would drve nto a wall before askng for nstructons around t! Arzona State Unversty, Department of Mathematcs and Statstcs 5 of 7

6 Example : More examples You are offered the opportunty to play a game where there are fourteen one-dollar blls, fvedollar blls, ten-dollar blls and one-hundred dollar bll n a bag. We buld a model frst. Bll Value $ $5 $0 $00 Totals Sum of the row Number 4 0 Probablty (decmal) Expected value component Bll Value Probablty The expected value s the sum of the contrbutons from each of the blls. A. What s the expected value for a sngle draw from the bag? $7.45 B. What s a far amount to pay to play ths game? $7.45 C. What s the expected value after 0 plays? 0 $7.45 = $74.50 D. What s the expected value after 00 plays? 00 $7.45 = $ E. Would you play ths game f you had to pay $5.00 for a chance? Of course! You expect to gan $ $5.00 =$.45 on average. F. For a $5.00 tcket, what s your expected value after 0 plays? 0 $.45 = $4.50 G. Would you play ths game f you had to pay $8.00 for a chance? Of course not! You expect to lose $0.55 on average. H. For an $8.00 tcket, what s your expected value after 0 plays? 0 $ = $ of 7 Arzona State Unversty, Department of Mathematcs and Statstcs

7 Example : You are offered the opportunty to play a game where there are 6 one-dollar blls, one fve-dollar bll, ten-dollar blls and twenty-dollar bll n a bag. To play you must put your own $5 bll n the bag. Do not answer ths queston based on the hgher probablty of the one's. It s totally dependent on expected values! Totals Bll Sze $ $5 $0 $0 Sum of the row Value n the game Number of blls n play 6 0 Probablty (decmal) Expected value component Game Value Probablty A. What s the expected value for a sngle draw from the bag? - $0.40 B. What s your expected value after 0 plays? 0 $0.0 = - $4.00 C. Would you play ths game f you expected to wn? No! In the next lessons, you look at random varables and a more complcated vew of mathematcal expectaton. Keep n mnd how smple the process really s as you wade through t. So far we have addressed the dscrete stuaton. We wll need the more demandng development to handle contnuous stuatons. One Last Example: Take a look at the odds stated on most lottery tckets. I found these on the web for one popular lottery. You mght clck on the lnk to get that explanaton at the bottom. Let s smplfy our stuaton just to look at wnnng the grand prze. For our demonstraton, let s set the prze at $00 mllon and assume you buy exactly one tcket. The expected value for that lofty goal s E = (00,000,000) + ( ) = Your expectaton for playng the lottery to wn the grand prze s negatve even wth a $00 mllon prze! Even factorng n all possble wnnng scenaros, you wll be a net loser. Otherwse, how can we guarantee that money wll be appled to schools and counselng for the people that never learned ths? Moral: Play t for fun! Play for a buck. Don t bet the farm! Arzona State Unversty, Department of Mathematcs and Statstcs 7 of 7