Management 3 Quantitative 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

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

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

Example:

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: