Expected Value Please note Before I slam you with the notation from Chapter 9 Section 2 I want you to understand how simple Mathematical Expectation really is My first simplification I will refer to it as Expected Value E from this point Objectives At the end of this lesson you should be able to 1 Define an expected value 2 Calculate expected value based on a probability model 3 Evaluate the wisdom of playing a game using expected value Background The other day I was confronted with a dose of reality My life expectancy is 74 years My wife has a 79 year expectancy Let s not get into whether men have shorter lives on average because they are married to women These are averages expectations expected values I doubt that my life will abruptly terminate on my 74th birthday Another grim dose of reality My insurance company thinks there is a possibility that I will be involved in a serious accident next month They expect me to be a negative cash flow to them when it happens Maybe I shouldn t drive next month Then again my chances of being injured while staying home are even greater That last sentence seems to link probability and expected value The situations above are parts of daily life They balance what could happen with what we expect to happen I could be hit by a meteor tonight I don t expect it to happen Meteorite activity on earth is relatively uniform today with an average of about one meteorite per year falling every 7 700 square miles 12 500 sq km There is a great deal of fuzziness in the various statements above First life expectancy does depend on genetic factors occupational factors and certainly could be influenced by plain dumb luck random factors like your spouse is a serial killer or a bad driver My likelihood of an accident depends on my age my vehicle s condition my driving habits and whether you are talking on your cell phone while driving near me My chances of getting wacked on the head by a meteorite increase dramatically during the Pleiades shower occurring from early June and to mid July They reach an hourly rate of 25 meteorites I ve watched them Not a single one hit me Let s get into the mathematics of expected values First let s do an exercise in calculating something very familiar to you your course grade Let s assume the grade is made of homework tests quizzes and a final exam Here is a typical weighting system found in many university syllabi Evaluation Homework Average Weight 40 Test Average 25 Quiz Average 15 Arizona State University Department of Mathematics and Statistics Final Exam 20 1 of 7 Expected Value Now suppose a student took three tests scoring 77 85 and 93 out 100 We all know the test average is 77 85 93 85 out of 100 So 85 is the average 3 Now the student also took 10 quizzes Each was worth ten points We recorded them in a table Quiz 1 2 3 4 5 6 7 8 9 10 Score 7 8 7 8 9 8 9 8 7 10 We can either add them up as we did in the test calculation or notice that we have repeated values Let s use the former to do the calculation This is sometimes called a weighted average So the quiz average is 3 7 4 8 2 9 1 10 81 8 1 out 10 10 10 There is a point of order here The student s average score on quizzes is 8 1 However we now have tests out of a 100 and quizzes out of 10 It would be smart to write both as a percent so they compare fairly The student has an 81 quiz result and an 85 test average Let s assume the student earned a 97 average on homework and a final exam grade of 89 So what is the final course grade Your gut instinct might be to add the four grades that would give an 88 average But you would be wrong We need to weight the components according to our syllabus standards The calculation follows 40 of Homework Average 25 of Test Average 15 of Quiz Average 20 of Final Exam Also remember we use percentages as a decimal value in multiplication So we get this 0 40 97 0 25 85 0 15 81 0 20 89 90 This is another weighted average Notice that by just averaging you overvalued the quiz result and undervalued the homework The student has a 90 average definitely an A grade You may wonder why we did all this Well now you know how to calculate your own grade when the professor uses weighted values in the syllabus And all the math we did here is what we will do in calculating an expected value Let s start with the tests Suppose that we change our view of the results The student has three grades of 75 85 and 93 Each has equal weight His probability for any of these grades is 1 in 3 His expected value for repeated testing is his arithmetic average if we make the reasonable assumption that the three tests don t change much 77 85 93 1 1 1 77 85 93 85 3 3 3 3 Now look at the quiz score average His probability of scoring a particular grade is not uniform The probability for a 7 is four in 10 The probability for a 9 is only two in 10 His expected value for repeated quizzing is his weighted quiz average 3 7 4 8 2 9 1 10 3 4 2 1 7 8 9 10 8 1 10 10 10 10 10 2 of 7 Arizona State University Department of Mathematics and Statistics Expected Value Finally for the course grade itself The value of each score to the student depends on how it is obtained whether in homework test quiz or final exam His expected value is again his weighted score just as before 0 40 97 0 25 85 0 15 81 0 20 89 90 Expected Value in Games Now let s apply this to a probability model Many times the model reflects a game we might want to play The expected value is a reliable way of removing perception from consideration Most people will agree to play a game with a 100 win if they only need to pay 1 The lottery proves this However expected value will tell exactly what it says What can you expect if you play repeatedly Example Suppose we are to choose from a bag with balls numbered 1 2 3 and 4 There is one of each Then we choose from another bag with balls numbered 3 4 and 5 The results are added together to create a point We ll call the two draws a turn The sample space has 4 3 12 outcomes possible but there are only 6 possible points on any turn 4 5 6 7 8 9 The lattice for this game is …