Management 3Quantitative Analysis

Winter 2015

Expected Value

Definitions:

• Expected indicates that we face at least two possible outcomes and that we have some information, estimates regarding probabilities, and values associated w/ those probabilities, on which to base an “expectation”.

• Value is measured in Dollars.

• Expected Value is an a priori formal construct. It is expressed in Dollars. It is not an actual result, but an expectation of the average actual result,

Example: Six-sided Die Toss

• What is the Expected Value of the Toss of a Six-sided Die?

This depends on how you define/associate Value in dollars to each of the possible outcomes: The #Dots = 1, 2, 3, 4, 5, 6If we receive one-dollar per Dot … then:

EV (die toss) =ƩPr(x)V(x), where #DotsEV = + +

EV = + +

EV = = $3.50

Definition: Expected Value

• The expected value of any uncertain situation ahead is the amount, on average, of money you gain(lose) each time you face that situation. We can think of the situation as a game, where there are two possible outcomes: win or lose.

• The expected value (in terms of a dichotomous game) is calculated as follows:

• EV = Pr(Win)V($Win) + [1-Pr(Win)]V($Lose)• EV = Pr(Win)V($Win) + Pr(Lose)V($Lose)

Example: Coin Toss

• Coin Toss- Dichotomous Chance event, i.e. - Pr(Heads) = ½ = Pr(Tails) = ½• Add money. You and I make a promise to pay

the winner $1 , i.e. you receive $1 from me if “Heads” and you pay me $1 if “Tails”. So, from your perspective:

EV (coin toss) =ƩPr(H)($1) + PR(T)(-$1)EV = ½ )EV = EV = $0

Definitions:

• If the expected value of a game is equal to zero “0”, then this is a fair game. Neither party has an advantage.

Advantages in games are a function of asymmetric probabilities or values.

What makes our coin toss “fair” is that the symmetry amongst probabilities and values. Each are equal for each player.

Example: Roulette

- The roulette wheel has 37 slots;- Slots are numbered 1-36 and colored red and

black alternatively, plus- One green slot numbered “00”.- We may place a $1 bet on any slot and, - Should the ball land on our chosen number, we

win $35.Is this a fair bet?

Example: Roulette

• Is Roulette a fair game?Only if the EV of the bet is = zero.Do the computation:EV (roulette) = EV = EV = we lose 2.7¢• Roulette is not a fair game

Example:

• Unfair Coin Toss- Say that the Dichotomous Chance events have different probabilities, i.e. - Pr(Heads) = and = Pr(Tails) = • You receive $1 from me if “Heads” and you pay

me $1 if “Tails”. From your perspective:

EV (coin toss) =ƩPr(H)($1) + PR(T)(-$1)EV = EV = EV = on average, you lose 32¢

Example:

• Changing the Values to translate asymmetric probabilities into a fair game:

Pr(Heads) = and = Pr(Tails) = How much should I pay you, when the coin lands “Heads” to make this a fair game? Set “EV” equal to “0” and substitute the variable X for the $1 in the first Value. EV = Solve for $X

Comparisons

• We can use Expected Value to compare courses of Action – separate, mutually exclusive, uncertain opportunities.

• When faced with two, or more, opportunities, a rational person would choose the opportunity with:

a) highest positive Expected Value, or b) least negative Expected Value.

Example: Two Chance Events

• You need some money, so you ask your dad. He says, let’s toss a coin and I’ll give you $20 if it comes-up Heads, but $0 if it comes-up Tails. You want to think about this, and while you do

• You ask your mother for some money. She says, let’s toss a die and I’ll give you $3 for each Dot that comes-up.

• You cannot do both. These are mutually exclusive opportunities – choosing one excludes the other.

Example, continued:

• Evaluate both offers using the Expected Value criterion.

EV (Dad) =

EV (Dad) =

Example, continued:

EV (Mom) = + +

Factor-out the EV = + + )EV = = $10.50