Expected ValueSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Expected ValueExpected ValueWhen faced with uncertainties, decisions are usually not based solely on probabilitiesA building contractor has to decide whether to bid on a construction job:20% chance of a $40,000 profit80% chance of a $9,000 lossDo we bid on the contract?Expected ValueThe chances for a profit are not very high but we stand to gain more than we stand to loseHow do we combine probabilities and consequences?Expected ValueConsider the following:A person aged 22 can expect to live 51 more yearsA married woman can expect to have 2.4 childrenA person can expect to eat 10.4 pounds of cheese and 324 eggs in a yearWhat do we mean we say “expect”?Expected ValueMathematical expectation can be interpreted as an averageA person aged 22 can expect to live an average of 51 more yearsA married woman can expect to have an average of 2.4 childrenA person can expect to eat an average of 10.4 pounds of cheese and 324 eggs in a yearExpected ValueEx: Suppose there are 1000 raffle tickets. There is a $500 prize for the winning ticket and a consolation prize of $1.00 for all other tickets. How much can a person expect to win playing the raffle?Expected ValueSol: Suppose all 1000 tickets are drawn and each person’s winnings was recorded. What would a person’s average winnings be?499.1100099911000150010001111500Winnings AverageNotice this is the probability of getting the winning ticketNotice this is the probability of not getting the winning ticketExpected ValueThe last slide tells us several things:Each amount won has a probability associated with itThe amount won is multiplied by its respective probabilityThe sum of the products is the expected valueExpected value is a weighted average (if we run the experiment many times, what is the average)Expected ValueWhat is a weighted average?Ex: A student computes his average grade in a course in which he took six exams: 75, 90, 75, 87, 75, and 90. He computes his average score as follows:8264926907587759075Expected ValueNotice he can also write the same average as:The average is the weighted average of the student’s grade, each grade being weighted by the probability the grade occurs 8262906187637562901873756907587759075Expected ValueOur raffle ticket example showed each amount had a probability associated with itWe did NOT consider the actual events but we associate numbers with the events that arose from the experimentExpected ValueA Random Variable assigns a numerical value to all possible outcomes of a random experimentEx: # of heads you get when you flip a coin twiceThe sum you get when you roll two diceExpected ValueEx. Consider tossing a coin 4 times. Let X be the number of heads. Findand . ,3,3 XPXP 2XPTTTTTTTHTTHTTTHHTHTTTHTHTHHTTHHHHTTTHTTHHTHTHTHHHHTTHHTHHHHTHHHHS,,,,,,,,,,,,,,,Expected ValueSoln. 1643 XP 1652 XPTTTTTTTHTTHTTTH HTHTTTHTHTHHTTHHHHTTTHTTHHTHTHTHHHHTTHHTHHHHTHHHHS,,,,,,,,,,,,,,, 16153 XPExpected ValueNote that the notation asks for the probability that the random variable represented by X is equal to a value represented by x.Remember that for n distinct outcomes for X, (The sum of all probabilities equals 1). xXP 11niixXPExpected ValueFormula for expected value (for n distinct outcomes: niiixXPxXE1Expected value of the random variable XExpected ValueEx. Find the expected value of X where X is the number of heads you get from 4 tosses. Assume the probability of getting heads is 0.5.Soln. First determine the possible outcomes. Then determine the probability of each. Next, take each value and multiply it by it’s respective probability. Finally, add these products.Expected ValuePossible outcomes:0, 1, 2, 3, or 4 headsProbability of each: 16116416616416143210XPXPXPXPXPTTTTTTTHTTHTTTHHTHTTTHTHTHHTTHHHHTTTHTTHHTHTHTHHHHTTHHTHHHH THHHHS,,,,,,,,,,,,,,,Expected ValueTake each value and multiply it by it’s respective probability:Add these products0 + 0.25 + 0.75 + 0.75 + 0.25 = 2 25.044475.033375.022225.01110000161164166164161XPXPXPXPXPExpected ValueEx. A state run monthly lottery can sell 100,000 tickets at $2 apiece. A ticket wins $1,000,000 with probability 0.0000005, $100 with probability 0.008, and $10 with probability 0.01. On average, how much can the state expect to profit from the lottery per month?Expected ValueSoln. State’s point of view:Earn: Pay: Net:$2 $1,000,000 -$999,998$2 $100 -$98$2 $10 -$8$2 $0 $2These are the possible values. Now find probabilitiesExpected ValueSoln. State’s point of view:We get the last probability since the sum of all probabilities must add to 1. 0000005.0998,999 XP 008.098 XP 01.08 XP 9819995.02 XPExpected ValueSoln. State’s point of view:Finally, add the products of the values and their probabilities 60.0$9819995.0201.08008.0980000005.0998,99922889898998,999998,999XPXPXPXPXEExpected ValueFocus on the Project:X: amount of money from a loan work outCompute the expected value for typical loan: 000,991,1$536.0000,250$464.0000,000,4$000,250$000,250$000,000,4$000,000,4$Failure FailureSuccess SuccessXPXPPPXEExpected ValueFocus on the Project:What does this tell us?Foreclosure:
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