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Unformatted text preview: MATH 221 Statistics for Decision - Making Lecture Notes Week 4 Mathematical Expectation / Probability Distributions Student Name Copyright 2010 by P.E.P. 1 Mathematical Expectation involves combining probabilities with consequences. Calculating the mathematical expectation of an event or sequence of events will help in the decision making process when there is uncertainty involved. Mathematical Expectation If the probabilities of obtaining the amounts a 1 , a 2 , . . . , or a k are p 1 , p 2 , . . . , and p k , where p 1 + p 2 + . . . + p k = 1 , then the mathematical expectation is E = a 1 p 1 + a 2 p 2 + . . . + a k p k In summation or Sigma Notation, this is written as E = a i p i In essence, to arrive at a mathematical expectation we merely multiply the given amounts, associated with the various events, times their respective probabilities and then sum these products. Example 1 To promote ticket sales, a theater owner gives 600 patrons a free chance in the drawing of a new stereo music system valued at $ 390 . What is the mathematical expectation of a patron who receives one of the chances? Solution Since there are only two events involved with this example, namely winning the stereo system or losing a chance to win, then the mathematical expectation becomes: E = ( amount to win ) ( probability of winning ) + ( amount to lose ) ( probability of losing ) or E = ( $ 390.00 ) ( 1 / 600 ) + ( $ 0.00 ) ( 599 / 600 ) and thus E = $ 0.65 + $ 0.00 = $ 0.65 Therefore, the mathematical expectation for a patron is $ 0.65 . Example 2 A stockbroker determines that the probabilities are 0.10 , 0.20 , 0.30 and 0.40 that the value of a stock will increase by $ 2.00, $ 1.50 and 75 cents, or not at all. What is the expected increase in the value of the stock? Solution Computing the expected increase in the value of the stock is the same as determining the mathematical expectation of the given amounts and their respective probabilities. E = $ 2.00 0.10 + $ 1.50 0.20 + $ 0.75 0.30 + $ 0.00 0.40 or E = $ 0.725 or 73 cents, rounded MATH 221 Statistics for Decision - Making Lecture Notes Week 4 Mathematical Expectation / Probability Distributions Student Name Copyright 2010 by P.E.P. 2 Decision Making Decision - making is that process which occurs when a choice must be made in the face of uncertainty. The choice usually will be made among two or more alternatives. By calculating the alternative with the highest mathematical expectation, the choice to make perhaps becomes " easy. " Example 3 A grab bag contains 10 packages worth $ 2 apiece, 5 packages worth $ 3 apiece and 10 packages worth $ 10 apiece. Is it rational to pay $ 5 for the privilege of selecting one of these packages at random?...
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