SLOT MACHINE MATH

Slot machines earned their name because they have a slot that accepts coins or tokens for play

from players. Most slot machines today are equipped with a currency acceptor box, or bill

validator, that allows the slot machine to accept currency. A player will insert coins or currency

into the machine to begin play. Most slots have two types of meters: hard meters and soft meters.

The two hard meters, which resemble the odometers in cars before the 1990s, track “coin-in” and

“coin-out,” respectively. Coin-in is the money inserted by the player into the slot machine. Coin-

out is the money that is paid by the slot machine to the player. Coin-in typically goes to the slot

machine hopper, a coin storage bank inside the machine through which all payouts are made. If

the hopper is full, the machine will divert additional coins to the drop bucket stored in a locked

cabinet beneath a slot machine. If the hopper becomes empty, a casino employee will undertake

a “fill” by adding coins or tokens. Slot meters record (1) total credits by coin-in and coin-out, (2)

total credit through the bill validator, (3) total credits, total credits played, total credits won, and

total credits paid, and (4) total games played, and total games won.

Handle is the total amount of money wagered or put at risk by a player. A player’s handle is

sometimes referred to as his action. In principle, handle may be calculated using the following

formula:

Handle = Pace × Average Bet × Duration

Pace refers to the speed of the game in decisions per hour in terms of hands dealt, wheel spins,

dice rolls, etc. Average bet refers to the average amount of money wagered per decision.

Duration is the length of time a player plays, in hours (a unit of time other than hours can be used

if the pace is also expressed in this unit). Slot machine meters and tracking systems allow for

easy determination of an individual player’s slot handle. For table games, however, it is
impractical to monitor every bet a player makes, or how long he plays, and the speed of the

game.

The is playing. Although technology may change this is the future, player table game handle is

currently estimated using reasonable values established through occasional observation of the

player’s average bet and duration of play. Although the speed of a table game is affected by

several factors, including the dealer and the number of players at the table, an average value is

often used for game pace.

Drop is the total amount of the contents in a game’s drop box (or drop bucket and currency

acceptor box in the case of slots). For table games, drop consists of the currency and foreign

chips in the drop box plus the value of credit instruments (markers) issued or redeemed at the

table. That is, drop = (cash + markers + chips) contained in the drop box.

For slots, the drop is the total amount of currency and coin in the slot machine currency acceptor

box and drop bucket. That is, drop = (coins + currency) contained in the drop bucket and

currency acceptor box. The slot drop does not include coins or tokens in the slot machine’s

hopper.

For table games, drop and handle are not equivalent. Handle, a theoretically precise notion, is

difficult to measure for table games. Drop, while empirically precise, is a conceptually elusive

term. Many factors can affect a table’s drop and it is not clear exactly what it is that drop

measures in terms of gambling volume.

Win refers to the total amount won by the casino at a table or device. For table games, win is the

amount of the drop minus the change in the table’s chip inventory, including chips issued during

fills and chips removed during credits.

Win = drop – missing chips = drop – (begin chips + fills – credits – end chips).
For slots, win is the amount in the drop bucket and bill validator less any jackpots paid and

accounting for any change in the hopper inventory, including fills.

Win = drop – jackpots – change in hopper inventory.

Theoretical win is the amount the casino expects to win and can be determined as follows:

Theoretical Win = H.A. x Handle

Since handle can be found by multiplying the average bet times the game pace times

the duration of play, theoretical win can be represented as:

Theoretical Win = H.A. x Pace x Avg. Bet x Duration

Theoretical win is also called expected win.

Win percentage is the win divided by handle:

Win Win Percentage = Win/Handle

As with the above description of win, this refers to the actual win percentage, and not theoretical

or expected win percentage. Actual win percentage represents the fraction of money wagered

that is retained by the casino over a given period or number of trials. The theoretical win

percentage is the house advantage, or the long-run average percentage of money wagered that is

retained by the casino. For many trials, the actual win percentage should approximate the

theoretical win percentage. For slots and other gaming devices, the actual win percentage is

known. For table games, however, win percentage is difficult to measure since handle is usually

unknown. For further discussion on these and other related issues, see the section later in this

chapter on different ways to express win rate. The percentage of money wagered that the casino
could expect to win in the long run is the theoretical, or expected, win percentage. This value is

just the house advantage:

Theoretical Win % = Theoretic Win/Handle = House Advantage

For many trials, the actual win percentage should be close to the theoretical win percentage.

Hold, or more specifically hold percentage, is the percentage of the drop that is won by the

casino. That is, Hold % = Win/Drop.

Note that for slot machines and other gaming devices, the hold percentage is equivalent to the

win percentage since drop is the same (in principle) as handle. Slot hold represents the

percentage of all money wagered by players that the machine retains.

As explained previously, accurately determining a player’s theoretical win (earning potential) for

table games is not possible without tracking every bet made by the player. In the case of high

rollers, however, the only efficient use of power may be to have a casino employee physically

observe and record every single bet placed by the premium player so that theoretical win can be

accurately assessed. This is because the amount of the rebate to high rollers can be significant.

For example, if a casino is willing to pay back 50% of theoretical loss, then the player would

receive 50% of the house advantage for each $1 wagered over the course of play. Thus, if the

edge were 1.2%, then 0.6% of each dollar wagered would be paid back to the player, regardless

of whether the player won or lost. The casino would retain the advantage at 50% of the usual

value, or 0.6%. When the player is a million-dollar player, this amount becomes worth tracking

by hand.

Slot machines with progressive meters do not have a set theoretical win in the same sense as

other slot machines. On non-progressive slot machines, the theoretical payout to players is the

same on every play because the opportunity to win, the winning combinations, and the amounts
of the payouts on winning combinations remain constant. On progressive machines, the

progressive meter increases with each

pull of the handle. For example, on a dollar progressive machine, a nickel of every dollar played

may be added to the progressive jackpot. Therefore, on any pull of the handle, the theoretical pay

back will increase as the progressive meter increases.

If no player wins the progressive jackpot, at some point, the increasing size of the jackpot creates

a theoretical payback that will result in a statistical advantage to the players. From this point to

the time a player wins the jackpot, the total amount of the paybacks to the players including the

progressive jackpot should exceed the total amount of money played by the players.

This slight statistical advantage is what attracts the slot teams. They find progressive jackpots

that have arisen to a level that favors the players. Only certain progressive slot machines will

meet the slot team’s criteria. First, the number of slot machines that are linked to the progressive

jackpot must be manageable. The team needs to monopolize all the slot machines to avoid the

risk that a non-member will win the jackpot. Second, the statistical frequency of hitting the

jackpot must be consistent with the slot team’s bankroll. No slot team has an unlimited bankroll.

Based on probabilities, the team needs to have enough cash on hand to play all the machines on

the carousel until one hit the progressive jackpot.77 For example, the previous chart shows a

progressive slot machine that has a cycle of 884,736 (and a probability of a little better than one

in one million of hitting the progressive jackpot). If the 77 The Poisson probability distribution

can be used to determine the probability of hitting the jackpot over the course of a specified

number of plays.

The probability of hitting the progressive jackpot is one in ten million, the slot team would need

an astronomical bankroll to have reasonable assurances that it would hit the jackpot before
running out of money. Therefore, slot teams avoid the “mega” jackpot progressive that have very

infrequent hits or winners. To minimize their potential exposure, they concentrate on progressive

carousels that feature lower jackpots with a higher frequency of payouts.