BRM Chapter 20 - The House Edge: Expected Value

Expected value - definition (words)

The average of all possible numerical outcomes, each outcome weighted by its probability of occurring.​

Expected value - formula

If outcomes are a1, a2, …, ak with probabilities p1, p2, …, pk, then expected value = a1p1 + a2p2 + … + akpk.​

Interpreting expected value

The expected value is the long‑run average outcome you would see if the random phenomenon were repeated many times.​

Example - Tri-State Daily Numbers "Straight" bet

You pay $0.50, choose a 3‑digit number; if the winning number matches exactly, you win $250, otherwise $0, with probability 1/1000 of winning.​

Straight bet - probability model

Outcomes: $0 with probability 0.999; $250 with probability 0.001.​

Example - StraightBox (six‑way) bet

You pay $1; you win $292 for an exact match and $42 if you have the same digits in any order; in the long run, $292 occurs once and $42 occurs five times per 1000 bets.​

StraightBox - probability model

Outcomes: $0 with probability 0.994; $42 with probability 0.005; $292 with probability 0.001.​

StraightBox - computing expected value

Expected value = (0)(0.994) + (42)(0.005) + (292)(0.001) = $0.502.​

Comparing Straight and StraightBox bets

The StraightBox bet is slightly better than the Straight bet because the state pays out slightly more than half of the money bet.​

Pari-mutuel system - definition

A system where all money bet on a game is pooled and a fixed proportion (such as one‑half) is paid out, divided among the winning tickets.​

Pari-mutuel example - New Jersey Pick 3

The state pools all bets on Pick 3 and pays out half the pool, equally shared among winning tickets; expected value depends on the total bet and number of winners.​

House edge - idea

The "house edge" is the positive expected value for the casino or lottery over many plays; the house designs games so its long‑run average profit is greater than zero.​

Expected value beyond gambling

Expected value also describes uncertain returns in contexts like investing in stocks or building a new factory.​

Example - vehicles per household

Using the distribution of vehicles per household (proportions of 0, 1, 2, …, 6 vehicles), the expected number of vehicles per household is 1.85.​

Law of large numbers - definition (words)

For a random phenomenon repeated many times independently, the mean of the actually observed outcomes approaches the expected value.​

Law of large numbers - interpretation

In many independent repetitions, the proportion of each outcome is close to its probability, and the average outcome is close to the expected value.​

Law of large numbers - gambling implication

Casinos are not "gambling": with enough bets, the casino's average winnings per bet are very close to the positive expected value, guaranteeing profit in the long run.​

Law of large numbers - insurance implication

Life insurance companies rely on known death probabilities; over many customers, average payouts are predictable, so premiums can be set to ensure profit.​

"How large is large?" issue

The law of large numbers does not specify how many trials are needed for the average to be close to the expected value; that depends on the variability of the outcomes.​

Variability and number of trials

The more variable the outcomes, the more repetitions are required for the average outcome to get close to the expected value.​

Lotto jackpots and variability

State lotteries with huge jackpots and tiny winning probabilities have extremely variable outcomes, so they require unrealistically many plays for the average to approach the expected value.​

Pari-mutuel and law of large numbers

Because lotto uses the pari-mutuel system, the state does not rely on the law of large numbers in the same way as casinos; payoffs depend on how much is bet.​

Player vs house - expected values

For gamblers, the expected value of winnings is negative; for the house, the expected value of winnings is positive, so gamblers as a group lose money over time.​

Uneven outcomes among gamblers

Although gamblers as a group lose, some individuals win big, some lose big, and some break even; this unpredictability for individuals is part of gambling's appeal.​

Betting systems - idea

Some gamblers use systems (like changing bet sizes based on previous outcomes) to try to overcome the house edge.​

Betting systems and independence

As long as successive plays are independent and the expected value per play is negative for the player, no betting system can change the long‑run average loss.​

Stronger law of large numbers for systems

A stronger version of the law of large numbers says that if you do not have infinite wealth and the game's trials are independent, your average winnings cannot be improved by any betting system.​

Statistics in summary - expected value

Expected value is the probability‑weighted average of all possible outcomes, used to measure the long‑run average result of a random phenomenon.​

Statistics in summary - law of large numbers

The law of large numbers guarantees that, in many repetitions, the mean outcome will eventually get close to the expected value.​

Statistics in summary - simulation to estimate EV

When outcome probabilities are unknown, simulation can be used to estimate both the probabilities and the expected value