Expected Value

 In gambling on an uncertain future, knowing the odds is only part of the story!  Example: I flip a fair coin. If it lands HEADS, you get. If it lands TAILS, you give me $1.  Do you take the bet?

Expected Value  All these scenarios carry the same sample space and same probabilities:  S = { H, T }, P(H) = P(T) = ½  But they do NOT all carry the same monetary outcome!

Expected Value  Knowing the odds is only half the battle, if different outcomes have different value to you.  Random Variable: A random variable is a function X which assigns a numerical value to each outcome in a sample space.  Example: If we decide to workout John Sander’s Loan, the sample space is: S = { success, failure } Each outcome has a different monetary value to Acadia Bank.

Expected Value   Let X be the amount of money we get back from a loan workout: X = $4,000,000 or X = $250,000  We are assigning a numerical value to each event in the sample space.  We’re describing each event with a number!

Expected Value  Describing events with a random variable: Let X = sum obtained from a roll of two dice. Let E be the event sum of the dice is greater than 8 Let F be the event the sum of the dice is between 3 and 6.  How do we describe the events E and F using our random variable X? E could be described as P(X > 8) F could be described as P(3 ≤ X ≤ 6)

Expected Value  What is P(X >8) = ?  What is P(3 ≤ X ≤ 6) = ?

Expected Value  Assigning a random variable to our probability space helps us balance risk with reward.  Fair-coin flipping example: Let X be our net profit from the game Reward exceeds risk. We should play! Risk equals reward. The game is fair Risk exceeds reward. We shouldn’t play!

Expected Value  What happens when the events are not equally probable?  S = { HEADS, TAILS} and P(H) = 0.25, P(T) = 0.75  What should the payoffs X be for this game to be fair?  So you should receive $3 for heads if you have to pay $1 for tails.

Expected Value  If we play this game 1000 times, how much can we expect to win or lose?  If we play a 1000 times: 250 heads, at $3 a piece means we receive $ tails, means we lose $1 so we pay -$750 After 1000 tosses, we net $0  This net is known as the expected value of the random variable X.

Expected Value  The expected value of any random variable is the average value we would expect it to have over a large number of experiments.  It is computed just like on the previous slide: for n distinct outcomes in an experiment!

Expected Value  Note that the notation asks for the probability that the random variable represented by X is equal to a value represented by x.  Remember that for n distinct outcomes for X, (The sum of all probabilities equals 1).

Expected Value  Ex. Consider tossing a coin 4 times. Let X be the number of heads. Find and.

Expected Value  Soln.

Expected Value  Ex. Find the expected value of X where X is the number of heads you get from 4 tosses. Assume the probability of getting heads is 0.5.  Soln. First determine the possible outcomes. Then determine the probability of each. Next, take each value and multiply it by it’s respective probability. Finally, add these products.

Expected Value  Possible outcomes: 0, 1, 2, 3, or 4 heads  Probability of each:

Expected Value  Take each value and multiply it by it’s respective probability:  Add these products = 2

Expected Value  Ex. A state run monthly lottery can sell 100,000 tickets at $2 apiece. A ticket wins $1,000,000 with probability , $100 with probability 0.008, and $10 with probability On average, how much can the state expect to profit from the lottery per month?

Expected Value  Soln. State’s point of view: Earn:Pay:Net: $2 $1,000,000 -$999,998 $2$100 -$98 $2$10 -$8 $2 $0 $2 These are the possible values. Now find probabilities

Expected Value  Soln. State’s point of view: We get the last probability since the sum of all probabilities must add to 1.

Expected Value  Soln. State’s point of view: Finally, add the products of the values and their probabilities

Expected Value  Focus on the Project: X: amount of money from a loan work out Compute the expected value for typical loan:

Expected Value  Focus on the Project: What does this tell us? Foreclosure: $2,100,000 Ave. loan work out: $1,991,000 Tentatively, we should foreclose. This doesn’t account for the specific characteristics of J. Sanders. However, this could reinforce or weaken our decision.