Expected Value

Central tendencies Mean Mode Median The most frequently used measure for describing central tendency is the expected value or average. Remember the values included in the average are “weighted” by the frequency.

Random Variable - A numerical variable whose value depends on the outcome of a chance experiment

Discrete Probability Distribution Gives the values associated with each possible x value Usually displayed in a table, but can be displayed with a histogram or formula

Discrete probability distributions 3)For every possible x value, 0 < P(x) < 1. 4) For all values of x, S P(x) = 1.

Suppose you toss 3 coins & record the number of heads. The random variable X defined as ... List the values of the Random variable. List the corresponding probabilities. The number of heads tossed X 0 1 2 3 P(X) .125 .375 .375 .125

Why does this not start at zero? Let x be the number of courses for which a randomly selected student at a certain university is registered. X 1 2 3 4 5 6 7 P(X) .02 .03 .09 ? .40 .16 .05 P(x = 4) = P(x < 4) = What is the probability that the student is registered for at least five courses? Why does this not start at zero? .25 .14 P(x > 5) = .61 .39

Expected Value is the mean weighted by its probability Expected Value is the mean weighted by its probability. Mean: Average Weighted Mean: each Random variable X multiplied by its probability Formulas for mean

What is the mean of this distribution? Let x be the number of courses for which a randomly selected student at a certain university is registered. X 1 2 3 4 5 6 7 P(X) .02 .03 .09 .25 .40 .16 .05 What is the mean of this distribution? m = 1(.02)+2(.03)+3(.09)+4(.25)+5(.40)+6(.16)+7(.05)

Setting up an expected value problem Each day a student puts $.50 in a candy machine to buy a $.35 pack of gum. She observes three possible outcome. 80% of the time she gets the gum and $.15 change. 16% of the time she gets the gum and no change. 4% of the time she gets the gum and the machine returns her $.50. Over a period of time, what is the cost of the gum?

DO NOT take the mean of three possible costs. 35,. 50 and 0. This = 28 DO NOT take the mean of three possible costs .35, .50 and 0. This = 28.3, and is NOT the expected value. The average cost (or expected value) takes into account that they DO NOT occur equally often.

Set up a discrete probabilty distribution WE emphasize that the term EXPECTED VALUE does not mean the value we expect in an everyday sense. In this gum example, the amount paid for the gum is never $.36. You would never expect to pay that, however, $.36 is what you would pay with a large number of purchases, or long period of time. This is the Expected Value. Set up a discrete probabilty distribution Cost (X) 0.35 0.50 P(x) 0.80 0.16 0.04 P(x) is the probability that the cost will occur. E(x) = (0.35)(0.80) +(0.50)(0.16)+(0)(0.04) = 0.36

Here’s a game: If a player rolls two dice and gets a sum of 2 or 12, he wins $20. If he gets a 7, he wins $5. The cost to roll the dice one time is $3. Is this game fair? A fair game is one where the cost to play EQUALS the expected value! X 0 5 20 P(X) 7/9 1/6 1/18 NO, since m = $1.944 which is less than it cost to play ($3).

Another way to look at the same problem In this case a fair game is one where the expected value =0! This is not a fair game. We can incorporate the cost of play into the game when we do the probability distribution. X (0-3) (5-3) (20 – 3) P(X) 7/9 1/6 1/18 E(x) = (-3)(7/9) + (2)(1/6)+(17)(1/18) = -1.06

Your turn An Auto repair shop’s records show that fifteen percent of the cars serviced need minor repairs averaging $100. Sixty-five percent need moderate repairs averaging $550, and twenty percent need major repairs averaging $1100. What is the expected cost of the repair of a car selected at random?

Your Turn A game involves throwing a pair of dice. The player will win, in dollars, the sum of the numbers thrown. How much should the organization charge to enter the game if they want to break even? If they want to average $1.50 per person profit.

Gambling affords a familiar illustration of the notion of an expected value. Consider Roulette. After bets are placed, the croupier spins the wheel and declares one of thirty –eight numbers, 00, 0, 1, 2….36 to be the winner. We will assume that each of these thirty-eight numbers is equally likely. Suppose we bet $1 on an odd number occurring.

Winning a dollar Losing a dollar Expected value