Chapter 20 The House Edge expected Values Chapter

Chapter 20 The House Edge: expected Values Chapter 15 1

Choose which one to play ? Multistate Lotteries Roulette Have enormous jackpots but very small probabilities of winning. Have a much more larger probabilities of winning but with small jackpots. Expected values provide one way to compare these two games. Chapter 15 2

The Tri-State Daily Numbers You pay $1 and choose a three-digit number. The state choose a three-digit winning number at random and pays you $500 if your number is chosen. Outcome : $0 $ 500 Probability 0. 999 0. 001 What are your average winnings? Chapter 15 3

The Tri-State Daily Numbers The ordinary average of the two possible outcomes $0 and $500 is $250. It makes no sense as the average winning because $500 is much less likely than $0. 500*0. 001 + 0*0. 999 = 0. 5 Chapter 15 4

Expected Values The expected value of a random phenomenon that has numerical outcomes is found by multiplying each outcome by its probability and then adding all the products. ai represents a possible amount, pi represents the probability of getting this amount Chapter 15 5

Buy a raffle A raffle has a first prize of $1000, a second prize of $500 and ten third prizes of $50 each. There are 500 tickets in all. Outcome Probability Amount A*P First Prize 1/500 = 0. 002 1000*0. 002=2 Second Prize 1/500 = 0. 002 500*0. 002=1 Third Prize 10/500 = 0. 02 50 50*0. 02 = 1 Total = 4 Chapter 15 6

The EV is average amount that you will win per play if you play the game over and over many times. Suppose that you buy all the tickets. Then you are guaranteed to win all the prizes. You win a total of $2000. That averages out to $2000/500 = $4 per ticket. Chapter 15 7

The EV is the fair price that you should pay to play the game If you pay $4 per ticket then you will exactly break even. If the organization that is putting on the lottery charges $5 per ticket, then they will make a profit (at the customers’ expense) of $1 per ticket, or $500 in all. Chapter 15 8

Deal or No Deal? (1) You choose one of four sealed cases; one contains $1, 000, and the others are empty. If you open your case, you have a 25% chance to win $1, 000 and a 75% chance of getting nothing (winning $0). (2) Or, you can sell your unopened case for $240, giving you a 100% chance of winning $240. • First option (open your case): EV=($1000)(0. 25) + ($0)(0. 75) = $250 • Second option (sell your case): EV=$240, no variation. • Make a Decision: Will you open or sell your case? Chapter 20 9

Deal or No Deal? (1) A 25% chance to win $1, 000 and a 75% chance of getting nothing. EV=$250 (2) A gift of $240, guaranteed. EV=$240 • If choosing for ONE trial: – option (1) will maximize potential gain ($1000) – option (1) will also minimize potential gain ($0) – option (2) guarantees a gain • If choosing for MANY (500? ) trials: – option (1) will maximize expected gain (will make more money in the long run) Chapter 20 10

The Law of Large Numbers If a random phenomenon with numerical outcomes is repeated many times independently, the mean of the actually observed outcomes approaches the Expected Value. Chapter 15 11

The Law of Large Numbers Gambling • The “house” in a gambling operation is not gambling at all. – the games are defined so that the gambler has a negative expected gain per play – each play is independent of previous plays, so the law of large numbers guarantees that the average winnings of a large number of customers will be close the (negative) expected value. No matter how you load the dice or stack the cards, As long as enough bets are placed, the Law of Large Numbers guarantees the profit of casino. Chapter 20 12