Expected Value

Expected Value Average expectation per game if the game is played many times Can be used to evaluate and compare alternatives in order to make decisions Expected value is used to estimate what we can expect to happen (theoretically)

Expected Value A game gives payoffs a1, a2, …, an with probabilities P1, P2, …, Pn. The expected value (or expectation) E of this game is In other words, multiply the payoff of each outcome by the chance of getting that outcome and add all these together.

Example A die is rolled and you receive $1 for each point that shows. What is your expectation? To calculate the expected value… What is the probability for each face of the die? What is the payoff for each face of the die?

Solution What is the probability for each face of the die? Each face of the die has probability 1/6 of showing What is the payoff for each face of the die? $1 for a 1, $2 for a 2, $3 for a 3, $4 for a 4, $5 for a 5, $6 for a 6

Solution Thus, the expected value is… This means that if you play this game many times, you will make, on average $3.50 per game.

Examples: A $20 bill, two $10 bills, three $5 bills and four $1 bills are placed in a bag. If a bill is chosen at random, what is the expected value for the amount chosen? How much should you charge to ensure you don’t lose money on this game? (1/10)*($20) + (2/10)*($10) + (3/10)*($5) + (4/10)*($1) = 59/10 = $5.90 At least $5.90 to break even.

Example In a game you flip a coin twice, and record the number of heads that occur. You get 10 points for 2 heads, zero points for 1 head, and 5 points for no heads. What is the expected value for the number of points you’ll win per turn? (1/4)*(10) + (2/4)*(0) + (1/4)*(5) = 15/4 = 3.75

Example In Monte Carlo, the game of roulette is played on a wheel with slots numbered 0,1,2…36. The wheel is spun and a ball is dropped. The ball is equally likely to land in any slot. To play, you bet $1 on any number other than zero. If the ball lands in your slot you get $36 (the $1 you bet plus $35). Find the expected value of this game.

Example There is an equally likely chance that a falling dart will land anywhere on the rug below. The following system is used to find the number of points the player wins. What is the expected value for the number of points won? Assume each piece is a square. Black = 0 points Gray = 20 points White = 40 points (6/15)*0 + (6/15)*20 + (3/15)*40 = 240/15 = 16 points

Your Turn… Find the expected value (or expectation) of the games described. Mike wins $2 if a coin toss shows heads and $1 if it shows tails. Jane wins $10 if a die roll shows a six, and she loses $1 otherwise. A coin is tossed twice. Albert wins $2 for each heads and must pay $1 for each tails.

Solutions Mike wins $2 if a coin toss shows heads and $1 if it shows tails $1.50 Jane wins $10 if a die roll shows a six, and she loses $1 otherwise $0.83 A coin is tossed twice. Albert wins $2 for each heads and must pay $1 for each tails. $1.00

Challenge Problem A dice game involves rolling 2 dice. If you roll a 2, 3, 4, 10, 11, or a 12 you win $5. If you roll a 5, 6, 7, 8, or 9 you lose $5. Find the expected value you win (or lose) per game. Hint: Of the 36 possible outcomes, how many different ways can you roll a 2, 3, 4, 10, 11, or 12? 1 way, 2 ways, 3 ways, 3 ways, 2 ways, 1 way (1/36)*$5 + (2/36)*$5 + (3/36)*$5 + (3/36)*$5 + (2/36)*$5 + (1/36)*$5 + (24/36)*(-$5) = -12/36 = -$0.33