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Chapter 3 Section 3.5 Expected Value Expected Value When the result of an experiment is one of several numbers, (sometimes called a random variable) we can calculate the expected value of the experiment. This is a measure of the numeric result with the probability of obtaining that result. This is the main idea behind many "risk-assessment" businesses such as insurance, stock trading, and gambling. Calculating Expected Value To find the expected value of an experiment that has a number associated with each result we take each number and multiply by the probability of getting that number and add them up. (Be careful to write your probabilities as decimals or fractions when you calculate expected value, do not use percentages! ) Result (A Number) n1 n2 n3 n4 n5 … nk Probability of getting Number p1 p2 p3 p4 p5 … pk Expected Value = n1p1 + n2p2 + n3p3 + n4p4 + n5p5 + … + nkpk We need to add up the number that results times the probability for all possible numbers that can result. Example If you spin the spinner to the right a person has agreed to pay you the amount shown in the color it lands on. What is the expected value for you to win playing this game? $ 10 $5 We begin by making a table of all possible numbers that can result along with their probabilities. Amount to win Probability of winning this amount (Percentage) Probability of winning this amount (Decimal) $ 40 $5 $ 10 $ 40 50% 25% 25% .5 .25 .25 Expected Value = (5)(.5) + (10)(.25) + (40)(.25) = 2.5 + 2.5 + 10 = 15 This means on average a person could expect to win 15 dollars each time they play this game. If they charge you $ 20 to spin this each time what is your expected value? Amount to win $5 $ 10 $ 40 $ -20 Probability of winning this amount (Decimal) .5 .25 .25 1 Expected Value = (5)(.5)+(10)(.25)+(40)(.25)+(-20)(1) = 2.5+2.5+10 – 20 = -5 Which means on average they you should expect to lose $ 5. Fair Games A game is considered fair if its expected value is zero. This means that the risk you are taking playing the game is equal to the amount you can expect to win. Suppose we play a game where you roll a regular die. If you roll an odd number (i.e. 1, 3, or 5) you get nothing, if you roll a 2 or a 4 you get $12, if you roll a 6 you get $60. What should be charged to make this game fair? We make a table again keeping in mind the 3 amounts you can "win"; 0, 12, 60. The cost of playing the game we will call x. The probability is 100% (or 1) that you will loose x. Amount to win Probability to win this amount (fraction) Expected Value is : Set equal to zero and solve : 14 x 0 14 x $0 $12 $60 $-x 3 6 2 6 1 6 1 3 2 1 0 12 60 x 1 0 4 10 x 14 x 6 6 6 This means that if you are charged $14 to play this game on average you and the person playing will break even. More than $14 and it will favor the person running the game, less than $14 and the game will be in your favor. Example The NFL has kept statistics on how many different teams a player will play for in a 5 year period, either 1, 2, 3, 4, 5, or 6. The table below shows the results. Number of teams played on Probability of playing on that number (percentage) Probability of playing on that number (decimal) a) What is the probability of playing on 6 teams? 1 2 3 4 5 6 40% 20% 10% 15% 10% 5% .4 .2 .1 .15 .1 .05 b) What is the expected number of teams a player will play on? 40%+20%+10%+15%+10%=95% (1)(.4)+(2)(.2)+(3)(.1)+(4)(.15)+(5)(.1)+(6)(.05) 100% - 95% = 5% .4 + .4 + .3 + .6 + .5 + .3 = 2.5 teams You have a 1, 5, 10, 20, and 50 dollar bill in your pocket. If you reach in without looking and pull one bill out at random what is the expected value? Bill 1 5 Probability of getting that that bill (fraction) 1 5 1 5 10 20 50 1 5 1 5 1 5 1 1 1 1 1 1 5 10 20 50 86 1 5 10 20 50 17.2 5 5 5 5 5 5 5 Decision Theory This can be used to reason numerically which is a correct decision to make by comparing expected values. Example A venture capital firm can invest in two different companies. Company A has a 40% chance of earning a $80,000 profit and a 60% chance of a $12,000 loss. Company B has a 70% chance of earning a $50,000 profit and a 30% chance of a $40,000 loss. Which is the better investment? To do this we compute the expected amount earned from each investment. Company A Company B Amount Earned 80,000 -12,000 Amount Earned 50,000 -40,000 Probability (percentage) 40% 60% Probability (percentage) 70% 30% Probability (decimal) .4 .6 Probability (decimal) .7 .3 Expected Value=.4(80000)+.6(-12000) Expected Value=.7(50000)+.3(-40000) = 24800 = 23000 Investing in Company A would be better it has an expected return of $24,000 compared to company B with an expected return of $23,000.
