15 SECTION 4.4: Expected Value This section tells you why most all gambling is a bad idea. And also why carnival or amusement park games are a bad idea. Random Variables Definition: Random Variable A random variable is a rule that assigns precisely one real number to each outcome of an experiment. Example 1: Random Variable for Grades Texas A&M University (and most all other universities) use letters to denote grades. Each letter grade also has a specific number attached to it, namely Example 2: Defining a Random Variable for Flipping Coins A fair coin is flipped three times. We are interested in keeping track of the number of heads in each trial. a) Associate a random variable with the outcomes to keep track of the number of heads, b) Give the probability distribution table for the random variable.
16 Histograms are a graphical way to represent how probability is distributed. To create a histogram: List the options for your random variable along the The represents probability For each random variable outcome x, over x draw a rectangle with a base of 1 unit and the height being equal to the probability P(X = x) of that random variable number Example 3: Histogram for Flipping Coins In Example 2, we let X be the random variable that assigned the to each outcome the number of times heads appeared. The probability distribution table was as follows: x (# heads) 0 1 2 3 P(X=x) 1/8 3/8 3/8 1/8 The corresponding histogram would be:
17 Example 4: Baseball Histogram Suppose that the number of singles, doubles, triples, homeruns, and no-hits were recorded for a certain baseball player for 465 times-at-bat. Let the random variable R be 0 for no-hits, 1 for a single, 2 for a double, 3 for a triple, and 4 for a homerun. a) Find the probability distribution for any given at-bat being a 0, 1, 2, 3, or 4, given the data below. Event No Hit Single Double Triple Homerun Frequency 271 112 33 3 37 R P(X = x) b) Create the histogram for the random variable R c) Use the histogram to find the probability that the player gets on base for any random at-bat
18 Expected Value Expected value is a special kind of average that tells us what to expect for an experiment over the long term. Definition: Expected Value Let X denote the random variable that has values x!, x!, x!,, x!, and let the associated probabilities be p!, p!, p!,, p!. (That is to say,,, and so on). Then the (or mean) of the random variable X, denote by, is given by: E X = x! p! + x! p! + x! p! + + x! p! Example 5: Calculating Expected Value Suppose X is a random variable whose values and probabilities are given by the following table: Random variable, x -2-1 0 1 2 3 P(X = x).2.15.05.35.15.10 a) Find the expected value, E(X). b) Find the expected value on your calculator.
19 Example 8: Expected Value A fair 6-sided die is tossed. Let X be the random variable assigning the number rolled to each outcome. Find E(X). Example 6: Expected Value of Coins One coin is taken at random from a bag containing 3 nickels, 4 dimes, and 7 quarters. Let X be the value of the coin selected. Find E(X). Example 7: Finding Random Variable & Expected Value of Game A game costs $3 to play. A fair 6-sided die is rolled. If you roll an even number, you win the amount of money equal to the number rolled. Otherwise, you win nothing. Find the expected net winnings for this game.
20 Example 8: Finding Insurance Premium A man wants to purchase a life insurance policy that will pay the beneficiary $30,000 in the event that the man s death occurs during the next year. The probability that the man will live another year is 0.96. What is the minimum amount the man will pay for his premium? (Hint: the minimum premium amount occurs when the life insurance company s expected profit is zero) Example 10: Probability Distribution 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0-2 -1 0 1 2 3 The probability distribution for a random variable X is given above. Find the following: a) P(X = 0) b) P(X 0) c) P( 1 < X 4) d) P(X 1)