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# INTR STATISTCL REASONING STAT 110

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This 3 page Class Notes was uploaded by Shane Marks on Monday October 26, 2015. The Class Notes belongs to STAT 110 at University of South Carolina - Columbia taught by L. Hendrix in Fall. Since its upload, it has received 27 views. For similar materials see /class/229650/stat-110-university-of-south-carolina-columbia in Statistics at University of South Carolina - Columbia.

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Date Created: 10/26/15

Chapter 20 Expected Value We have been studying how to nd the probability ofa speci c event When we buy a raf e ticket we are usually curious what the probability ofa win is but we are also interested in determining how much we will win on average Example 1 Suppose a sorority is selling 1000 raf e tickets for 100 each One ticket will be drawn at random and the winner receives 20000 There are two possible outcomes here 1 Win 20000 so net gain is 199 with probability 11000 2 Do not win so net gain is 1 with probability 9991000 80 how would we find the average gain Definition The expected value of a random phenomenon that has numerical outcomes is found by multiplying each outcome by its probability and then adding all the products In symbols ifthe possible outcomes are a1 a2 a3 ak and their probabilities are p1 p2 p3 pk then expected value a1p1 a2p2 a3p3 akpk Notation We call E the expected value operator We read Egain as expected gain Note An expected value is an average of all the possible outcomes Example 1 cont d For the sorority fundraiser example find the expected gain noticing the use ofthe de nition of expected value this time Chapter 20 Page 1 Example 2 Consider a lottery game where you pay 050 and choose a three digit number The lottery picks a three digit number randomly and pays you 250 ifyour number matches What is the probability model for your gain What is your longrun average gain That is nd your expected gain Example 3 An insurance company sells a particular policy that costs the insured 1000 to purchase The insurance company knows that 1100 ofthese policy holders will le a claim worth 20000 1200 will le a 50000 claim and 1500 will le a claim of 100000 Find the insurance company s expected gain for this policy Clicker Example Suppose we have the following distribution ofoutcomes for a loaded die Outcome 1 I2 I3 I4 5 I6 I Probability 05 01 01 01 01 01 Find the expected number you ll get when rolling this die A 1 B 15 C2 D25 E3 Example 4 Use simulation to estimate the expected number of children a couple will have ifthey follow the scheme that they will have children until they have a girl or until they have three children Recall the probability ofa girl is 049 and a boy is 051 Chapter 20 Page 2 Thinking about Expected Value as a Longrun Average Notice that the expected value ofa variable is a weighted average of all the possible outcomes And it represents the longrun average we will actually see ifwe observe the random phenomenon many times It makes sense intuitively to believe that the more trials we observe the closer our observed average should be to the true average This can also be proved mathematically and this result is known as the Law of Large Numbers Law of Large Numbers lfa random phenomenon with numerical outcomes is repeated many times independently the mean ofthe observed outcomes approaches the expected value The Law of Large Numbers is closely related to the idea of probability in many independent repetitions of a random outcome the proportion oftimes a random outcome occurs approaches the probability ofthat random outcome and the average outcome obtained will approach the expected value as you increase the number of observations The Law of Large Numbers explains why gambling is a business for a casino they use the fact that the average winnings ofa large number of customers will be quite close to its expected value and that is how they make money The more variable a random outcome is the more trials are necessary for the mean of the observations to be close to the expected value Chapter 20 Page 3

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