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Expected Value

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When faced with uncertainties, decisions are usually not based solely on probabilities A building contractor has to decide whether to bid on a construction job: 20% chance of a $40,000 profit 80% chance of a $9,000 loss Do we bid on the contract?

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Expected Value The chances for a profit are not very high but we stand to gain more than we stand to lose How do we combine probabilities and consequences?

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Expected Value Consider the following: A person aged 22 can expect to live 51 more years A married woman can expect to have 2.4 children A person can expect to eat 10.4 pounds of cheese and 324 eggs in a year What do we mean we say expect?

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Expected Value Mathematical expectation can be interpreted as an average A person aged 22 can expect to live an average of 51 more years A married woman can expect to have an average of 2.4 children A person can expect to eat an average of 10.4 pounds of cheese and 324 eggs in a year

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Expected Value Ex: 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?

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Expected Value Sol: Suppose all 1000 tickets are drawn and each persons winnings was recorded. What would a persons average winnings be? Notice this is the probability of getting the winning ticket Notice this is the probability of not getting the winning ticket

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Expected Value The last slide tells us several things: Each amount won has a probability associated with it The amount won is multiplied by its respective probability The sum of the products is the expected value Expected value is a weighted average (if we run the experiment many times, what is the average)

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Expected Value What 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:

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Expected Value Notice he can also write the same average as: The average is the weighted average of the students grade, each grade being weighted by the probability the grade occurs

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Expected Value Our raffle ticket example showed each amount had a probability associated with it We did NOT consider the actual events but we associate numbers with the events that arose from the experiment

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Expected Value A Random Variable assigns a numerical value to all possible outcomes of a random experiment Ex: # of heads you get when you flip a coin twice The sum you get when you roll two dice

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Expected Value Ex. Consider tossing a coin 4 times. Let X be the number of heads. Find and.

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Expected Value Soln.

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Expected Value Note 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).

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Expected Value Formula for expected value (for n distinct outcomes: Expected value of the random variable X

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Expected Value Ex. 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 its respective probability. Finally, add these products.

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Expected Value Possible outcomes: 0, 1, 2, 3, or 4 heads Probability of each:

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Expected Value Take each value and multiply it by its respective probability: Add these products 0 + 0.25 + 0.75 + 0.75 + 0.25 = 2

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Expected Value Ex. 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?

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Expected Value Soln. States point of view: Earn:Pay:Net: $2 $1,000,000 -$999,998 $2$100 -$98 $2$10 -$8 $2 $0 $2 These are the possible values. Now find probabilities

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Expected Value Soln. States point of view: We get the last probability since the sum of all probabilities must add to 1.

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Expected Value Soln. States point of view: Finally, add the products of the values and their probabilities

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Expected Value Focus on the Project: X: amount of money from a loan work out Compute the expected value for typical loan:

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Expected Value Focus on the Project: What does this tell us? Foreclosure: $2,100,000 Ave. loan work out: $1,991,000 Tentatively, we should foreclose. This doesnt account for the specific characteristics of J. Sanders. However, this could reinforce or weaken our decision.

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