Learning Goals: * Determining the expected value from a probability distribution. * Applying the expected value formula to solve problems.

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1 Learning Goals: * Determining the expected value from a probability distribution. * Applying the expected value formula to solve problems. The following are marks from assignments and tests in a math class. What could this person receive as a final mark in the course? Find the average:

2 Lets see it another way... from a probability perspective This would be the mark to EXPECT Expected Value Defined Is a discrete random variable, with some probability distribution, is sampled repeatedly, the limiting value of the average is called "the expected value". The quantity that you can expect to obtain when an experiment is performed A weighted average!

3 The Formula Let X represent a discrete random variable with the probability distribution function P(X). Then the expected value of X denoted by E(X), or μ, is defined as: To calculate this, we multiply each possible value of the variable by its probability, then add the results. E(x)= { x 1 P(X=x 1 )} + { x 2 P(X=x 2 )} + { x 3 P(X=x 3 )} { x n P(X=x n )} E(X) is also called the mean of the probability distribution.

4 Let the random variable X be the winnings associated with each roll. From the table, the winnings range from $5.00 to $10.00, so X may take the values 5, 6, 7, 8, 9, or 10. You can obtain the probabilities with the aid of the tabulated rolls of two dice. Winnings (x) Sum Probabilities p(x) xp(x) 5 7 6/36 5 *6/ or 8 5/36 + 5/36 6 *10/ or 9 4/36 + 4/36 7 * 8/ or 10 3/36 + 3/36 8 * 6/ or 11 2/36 + 2/36 9 *4/ or 12 1/36 + 1/36 10 * 2/36 Sum 36/36 250/36 = 6.94 Expected Value If you pay $7.50 per game, then you would expect to lose $7.50 $6.94 = $0.56 per game in the long run. This would be a fair game only if the cost to play was $6.94. Example 2: The following table is a relative frequency distribution showing the number of first, seconds, thirds, fourths, and fifth place finishes Jane earned during 50 swimming races The expected value tells us that Jane s expected finish for a race is 1.78 place. Obviously, this will never be the result of any one particular race, rather it represents the average finish of many races. The sum of the expected values does not have to equal one of the values for the random variable.

5 Example Three people are chosen from a group consisting of 4 men and 3 women. a) Determine the probability of the chosen committee having at least one man on it. b) Determine the expected value of choosing a man. Solution: a) Solution 1 Solution 2 b) Example 3: Consider a simple game in which you roll a single die. If you roll an even number, you gain that number of points, and if you roll an odd number, you lose that number of points. a) Show the probability distribution of points in this game. b) What is the expected number of points per roll? c) Is this a fair game. Solution:

6 In groups you will present your answers to the class for the following questions: 1. The manager of a telemarketing firm conducted a time study to analyze the length of time his employees spent engaged in a typical sales-related phone call. The results are shown in the table below, where time has been rounded to the nearest minute. a) Define the random variable X. b) Create a probability distribution for these data. c) Determine the expected length of a typical sales-related call. Time (min) Frequency A drawer contains four red socks and two blue socks. Three socks are drawn from the drawer without replacement. a) Create a probability distribution in which the random variable represents the number of red socks. b) Determine the expected number of red socks if three are drawn from the drawer without replacement. 3. Find a set of data that meets the following two conditions: * the data are represented by a frequency table. * a discrete random variable can be used to represent the outcomes. a) Create a probability distribution for your data set. b) Use your data set to determine the expected value. 4. The graph below shows the probabilities of a variable, N, for the values of N = 0 to N = 4. Is this the graph of a valid probability distribution? Explain. P(N = n) n