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# Class Note for MATH 1313 with Professor Flagg at UH 2

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Date Created: 02/06/15

Math 1313 Section 82 Expected Value You already understand the concept of the average of a set of numbers Sometimes we want to know the average of a set of data but we only have the probability distribution available not the raw data In this case we can sill nd the average but we give it a new name expected value and nd it using a different approach De nition The mean or average of a set of numbers n1 n2 n3 n is n1 n2 n3n r The mean is one measure of central tendency or the center of a group of data When we have de ned a random variable and are working with its probability distribution we talk about its expected value rather than its mean or average De nition LetX denote a random variable that assumes values x1 x2 x with associated probabilities p1 p2 p respectively Then the expected value of X denoted EX is givenby EX xlp1 x2p2 xnpn Example 1 Find the expected value of X given its probability distribution x 0 1 2 3 4 5 PXx 125 25 1875 25 0625 125 Math 1313 Class Notes 7 Section 82 Page 1 of4 Example 2 The owner of a newsstand estimates that the weekly demand for a certain magazine are as follows Demand 10 11 12 13 14 15 Probability 05 15 25 30 20 05 Find the number of magazines he should expect to be demanded per week Example 3 A group of private investors intends to purchase a hotel currently being offered for sale in a resort city Records obtained from the hotel indicate the occupancy rates with corresponding probabilities during the May 7 September tourist season and are shown in the following table Magic Carpet Hotel Occupancy Rate 75 80 85 90 95 100 Probability 35 21 18 15 09 02 a Find the expected occupancy rate for the Magic Carpet Hotel b If the Magic Carpet Hotel has 175 rooms how many rooms can the investors expect to be occupied on a randomly selected night during the tourist season c The Magic Carpet Hotel charges 125 per night per room What is the expected income on a randomly selected night in tourist season Math 1313 Class Notes 7 Section 82 Page 2 of4 Example 4 A bag contains 30 quarters 25 dimes 45 nickels and 50 pennies A coin is drawn at random from the bag What is the expected value of the coin Odds Odds and probability are not the same thing although they are de nitely related We speak of odds in two ways either the odds in favor of an event occurring or the odds against an event occurring Odds are often expressed as a ratio of two numbers such as a b or a to b Here are some formulas which are helpful in solving odds problems Odds in favor of an event If PE is the probability of an event E occurring PE PE PE 1 1 PE PE5 then the odds in favor of E are Odds against an event If PE is the probability of an eventE occurring then 1 PE PE the odds againstE are P E P E PE 2 0 Probability of an event given the odds If the odds in favor of an eventE a abi occurring are a to b the then probability of event E occurring is PE Math 1313 Class Notes 7 Section 82 Page 3 of4 We ll use the third formula extensively Example 5 The probability that the Texans will win the Super Bowl is 002 What are the odds in favor of the Texans winning the Super Bowl Example 6 The odds that it will rain on Thursday are 3 to 5 What is the subjective probability that it will rain What is the probability that it will not rain Example 7 The odds against Shelly winning a drawing are 991 What is the probability that Shelly will win the drawing Math 1313 Class Notes 7 Section 82 Page 4 of4

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