Gambling tax report example -- Mark C. Nicely, gambling tax expert witness

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EXPERT WITNESS REPORT OF MARK C. NICELY, MATHEMATICIAN

AND CASINO GAMING CONSULTANT

I. QUALIFICATIONS.

I am trained as a mathematician and computer engineer, having graduated with a

Bachelors of Science degree in Electrical and Computer Systems Engineering from Renssalear

Polytechnic Institute. I am currently a director of gaming design for a game division of

International Game Technology (IGT), the world’s largest casino slot game and casino systems

manufacturing company. I have extensive experience in the casino and gaming industry, as

reflected in my curriculum vitae attached as Exhibit 1.

II. ENGAGEMENT AND SUMMARY OF OPINIONS.

Casino attached as Exhibit 2 and (B) calculate and explain the odds that the

Petitioners could have won more than their total gambling losses for a given year from any

amount of untracked gambling (i.e. from profits made solely from machine-paid awards whilst

playing without a Player Club Card.)

It is my opinion, as discussed more fully below, that for the years 2004 and 2005, the

odds are astronomically unlikely that the Petitioners achieved overall net winnings from

gambling in either year.

Petitioners,

I, Mark C. Nicely, hereby provide the following expert witness report on behalf of

I have been retained by Petitioners

, through their counsel, to

(A) explain the data listed in the Acres Advantages

Patron Management report from

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III. DISCUSSION OF PLAYER TRACKING REPORT.

A. General Description of the Player Tracking.

One of the primary purposes of a player tracking system such as the Acres Advantage

System, is to collect, store and report data captured from any game transaction on a slot machine

or video poker machine when a Player Club Card is active. When a player inserts his/her Player

Club Card into a given machine, there is usually one or more visual indicators to the player as to

whether the card is active. Most player tracking devices have an LED or LCD display that can

display messages to the player, such as:

Insert Player Card

Re-Insert Player Card

Card Accepted – Welcome <player name>

You currently have XXXX points

Most player tracking devices also have a lighted boarder or bezel surrounding the Player

Club Card insertion slot. The bezel displays a red color if no Player Club Card is inserted or if an

inserted Player Club Card is not properly read. If a Player Club Card is inserted and has been

properly read, the bezel displays a green color. Just as one may have to reswipe a credit card one

or more times in order for the credit card to be correctly read by a credit card reader, a given slot

machine Player Club Card reader may require a player to reinsert his/her Player Club Card one

or more additional times until said Player Club Card has been properly read.

When a player’s Player Club Card has been correctly read, every game subsequently

played while that Player Club Card remains in the Player Club Card reader is associated with that

player’s player tracking account. When a Player Club Card is removed from a Player Club Card

reader, the Player Club Card is unregistered from that machine.

-3-

If a player does not insert his/her Player Club Card into a given machine or if the

machine does not correctly read the Player Club Card and the player does not reinsert the Player

Club Card until a valid card read has occurred, then any play on that machine by that player will

not be associated with that player’s Player Club Card account.

The primary reason most casinos purchase and operate player tracking systems such as

the Acres Advantage System is to support a player loyalty program wherein a player’s tracked

machine play (i.e. gambling on a slot machine or video poker machine whilst using his/her

Player Club Card) can earn him/her possible perquisites such as premium parking, shorter VIP

lines for buffet, special room rates, special newsletter, invitations to special events such as slot

tournament, gifts such as show tickets, shirts, hats, etc., entries into special drawings, free play,

and/or other rewards. In order to accurately and fairly ascertain what reward a given player

should be offered, a player tracking system is used to record actual play. Relative to the player

loyalty program rules established by a given casino, a player earns another loyalty point for

every n dollars of play on a certain type of machine. For example, at some casinos, a player

earns 1 loyalty point for every $1 wagered during player-tracked play on slot game and earns 1

point for every $2 wagered during player-tracked play on a video poker machine.

Many player tracking systems can and do track estimated or actual player table game

wagers. Since no player tracking information for table games is provided in the report in Exhibit

2, I will not discuss the corresponding table game player tracking operational details, which are

different and more complex than with the tracking of slot machine and video poker machine

play.

-4-

Player tracking systems can also be used for optimizing casino operations such as

determining the effectiveness of mailers, promotions and other marketing campaigns,

understanding when, how and where higher value players play, etc.

A non-gambling equivalent of a player tracking system would be a frequent flier program

offered by most airlines. A given airline customer gets loyalty program credit for each flight

completed with a valid frequent flier number just like a given casino player gets loyalty program

credit for each play session completed with a valid Player Club Card. Conventional wisdom

indicates that airline frequent flier tracking systems are basically accurate, excluding occasional

problems due to error made during hand-entry of a frequent flier number by a customer or agent.

While casino player tracking systems of slot machine play have the same inherent systemic

accuracy as airline frequent flier systems, they enjoy even higher effective accuracy as there are

not equivalent occasions for humans to enter in player club numbers by hand, nor are there

complexities such as refunded or upgraded tickets, partially completed itineraries, etc.

Specifically, I have never heard of any problems or read of any accuracy issues over the

years regarding player tracking of slot machine and video poker machine play in any of the

major industry trade press to which I subscribe (Casino Slot Player, Strictly Slots, International

Gaming and Wagering Business). [Note: there have been numerous press articles regarding

player disputes in the area of table game tracking when casino staff estimation is required to

perform the tracking, but such issues are not germane to the issue at hand.]

Frequent flier programs and player tracking programs are widely known to be accurate

because there are so many users (be it fliers or gamblers) continually checking the accuracy of

their loyalty point totals; any systemic problem in any such a system would be exceedingly

unlikely to go unnoticed and unreported.

-5-

B. Slot Statistics Entries on the Acres Advantage Patron Management Report.

The player tracking report included in Exhibit 2 is a report that has the title “Acres

report is consistent with the output of an Acres Advantage System. The Acres Advantage System

was developed and sold by Acres Gaming, Inc. Acres Gaming, Inc. was acquired in 2003, by the

company for which I presently work: International Game Technology (IGT). IGT has taken over

the product support responsibilities for Acres Advantage Systems, including the specific Acres

Advantage Systems that generated the report in Exhibit 2.

Definition of terms used in Slot Statistics

The value of each is expressed in US dollars, for the identified player, for the listed

period of time:

Coin In: Total actual wagers placed

Coin Out: Total actual winnings paid by machine

Jackpots: Total actual winnings paid by hand, such as when W-2G is required

Act Win: Actual casino profit (loss)

Theo Win: Average mathematically expected casino profit relative to actual wagers

Free Cash Play Used: The nominal wager value of free game play provided to the player

by the casino

Additional information:

The phrase “wagers placed” means money that is irreversibly committed to the casino.

For example, when I insert a $20 bill into a slot machine or a video poker machine in the United

States, this is always a reversible action. I am not committed to having to bet all or even some of

Advantage Patron Management – [Player Workbook –

”. This

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this money. I can change my mind, press a CASH OUT button and receive the equivalent of my

$20 back (though it may be in a different form, such as in coins, tokens, redeemable voucher).

Only when I activate a gambling proposition for a specific amount of money, for example, one

spin of the reels in a slot machine for $5, have I irreversibly placed a wager.

The phrase “paid by hand” means that a player receives payment from a casino staff

member, in lieu of the slot machine or video poker machine having done so. An award paid by

hand (also known as a “hand pay”) could involve any of the forms of money that a machine

might dispense (as listed above), and/or could involve a form of money which typically comes

only from human action, such as a check, cashier’s check, money-order, etc.

However, the process of a player redeeming tokens, vouchers, etc. for cash money is not

considered to be, and is not referred to in casino parlance as, a “paid by hand” or “hand pay”

event.

Slot machines and video poker machines typically have a programmable setting for the

award amount threshold for triggering a hand pay. For machines to be placed in the United

States, the threshold amount is set at the factory to be $1,200, consistent with the W-2G

threshold. This assures that the required W-2G award procedures and paperwork are properly

completed before the corresponding winnings are given to the player.

On many slot machines and video poker machines, the operator is able to change the

hand-pay threshold settings. Though I am aware of some operators that have lowered the

jackpot-pay threshold setting to an amount lower than $1,200, I have only seen this done for

some coin-based, low-denomination, low-stakes machines. A prime example of such a device

would be coin-dispensing, electro-mechanical, nickel denominated slot machines with a max bet

of either 15 cents (three nickel bet on the single available payline) or 25 cents (one nickel bet on

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each of five available paylines) which were relatively prevalent in Nevada, Atlantic City and

Mississippi until they started to be displaced by multi-coin/multi-line video-based slot machines

starting around 10 years ago.

The main reasons why some casinos used to set such low-denomination, low-stake, coin-

dispensing machines, discussed next, are not really applicable to higher denomination devices

($1 or higher), or to high-stake devices (on the order of $1 or higher), and are certainly not

applicable to Ticket-In / Ticket-Out devices which dispense only vouchers and which neither

accept nor dispense coins.

One main reason that a casino might lower the jackpot-pay threshold on a coin-based,

low-stake, nickel machine relates to hopper fills. Before the Ticket-In/Ticket-Out technology,

which prevails on present-day casino floors in the United States, slot machines accepted coins as

well as notes and dispensed coins. The coins available for dispensing sit in a storage area known

as a hopper. If the required number of coins are not available to completely fulfill a player’s cash

out request, the machine locks up and displays a message such as “CALL ATTENDANT”. For

the player’s cash out request to complete, a casino staff member (usually two staff members,

based upon the given casino’s security policy) must open the machine and refill the hopper with

coins. This typically takes quite a few minutes to accomplish after the appropriate casino

personnel arrive, which in itself can take an indefinite amount of time. In my experience, trying

to get the attention of casino floor staff, in a noisy environment, without going so far from my

machine that I risk some other patron trying to sit at my machine, is extremely frustrating.

Some casinos used to set the jackpot threshold on a coin-based, low-stake nickel machine

at a sub W-2G threshold, such as $500 or $250, for the following reason. Were the player to try

to immediately cash out a $500 win, this would require 10,000 coins and this would most

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certainly cause a hopper-fill machine lock-up event with all of the corresponding negative

customer consequences. Even if the player continues playing on the same machine, with a

maximum wagering of 15 cents or 25 cents per play, he/she is unlikely to whittle down his/her

balance significantly. By setting the jackpot threshold to the top award value for a low-stake

nickel machine, the casino can exchange a certain-to-be annoying hopper-fill hand-pay

experience with a jackpot-win hand-pay experience. The latter is expected to be more pleasant

for the player for a few reasons. A jackpot-win lock up is typically accompanied by a rather loud

jackpot bell and flashing lights. The player can “bask in fame” as the clearly identified big

winner amongst nearby patrons. (This speaks to the other reason that casinos will do this – to

create an environment of excitement. It is conceivable that a casino might set a $1 denominated,

$2 max bet, enormous Big-Bertha style device with a low jackpot-pay threshold for the spectacle

it creates.) When a player cashes out, he/she was intending to leave that machine, possibly for

some other activity. However, a player encountering a jackpot-win lock-up was likely not

planning on leaving, so remaining at the machine is not as drastic of a change of plans. The

audio/visual commotion is likely to lead to the arrival of casino staff personnel much more

quickly than in a hopper-fill situation and likewise can be resolved much faster: it takes much

less time to pay the player $500 in notes and reset the machine (usually without having to open

the machine), relative to the significant amount of time needed to refill a hopper.

However, I have never heard of a casino setting a sub $1,200 jackpot-pay threshold on a

high-denomination ($1 and above) slot machine, or high-stake ($1 to $2 or higher) slot machine.

High denominated devices, such as denominations of $1, $5, $25 and $100 are often specially

designed to offer awards such as $1,000 or $1,199 to expressly allow players to enjoy big wins

without having to suffer through a W-2G lock-up. Indeed, the Petitioner,

-9-

confirms that she never experienced a machine lock-up for an award less than $1,200. Therefore,

any values listed in the Jackpot column in Exhibit 2, I recognize as coming from a W-2G

recorded award.

The interrelationship of the first four primary statistics can be described as follows:

(1) Coin In = Coin Out + Jackpots + Act Win

Or conversely:

(2) Coin In – (Coin Out + Jackpots) = Act Win

This can also be expressed in a more verbal fashion:

Casino profit is total player wagers minus awards paid to player. Awards paid to

player include awards directly paid by the slot machine as well as awards

handed to the player by casino staff.

The above listed terms are all well-known and much-used terms within the gambling

industry. For example, the terms “Coin In”, “Coin Out” and “Jackpots” can all be found

throughout the Nevada’s Gaming Statutes and Regulation, such as the following section of the

regulations from the Nevada Gaming Commission and State Control Board Web site:

http://gaming.nv.gov/stats_regs.htm. Please note: I have highlighted and underlined the terms–

these highlights and underlines are not in the original.

2.040 Meters for conventional gaming devices, system supported and system based

games.

1. All gaming devices must be equipped with electronic digital storage meters of at least 10

digits capable of displaying the information listed in this section on demand. These meters,

listed below, must accumulate the following information in units equal to the denomination of

the device or in dollars and cents. Devices configured for multi-denomination play must

display the required information in dollars and cents.

(a) Coin In. The machine must have a meter specifically labeled “Coin In” that accumulates

the total value of all wagers, whether the wagered amount results from the insertion of coins,

tokens, currency, deduction from a credit meter or any other means. This meter shall:

-10-

(1) Not include subsequent wagers of intermediate winnings accumulated during game play

sequence such as those acquired from “double up” games;

(2) For multi-game and multi-denomination/multi-game gaming devices, provide the

information necessary, on a per paytable basis, to calculate a weighted average theoretical

payback percentage; and

(3) For gaming devices which are considered slot machines and which contain paytables

with a difference in theoretical payback percentage which exceeds 4 percent between wager

categories, maintain and display coin in meters and the associated theoretical payback

percentage, for each wager category with a different theoretical payback percentage, and

calculate a weighted average theoretical payback percentage for that paytable;

(b) Coin Out. The machine must have a meter specifically labeled “Coin Out” that

accumulates the total value of all amounts directly paid by the machine as a result of winning

wagers, whether the payout is made from the hopper, to a credit meter or by any other

means. This meter will not record amounts awarded as the result of an external bonusing

system or a progressive payout;

accumulates the total value of coins or tokens diverted to the drop;

(d) Attendant Paid Jackpots. The machine must have a meter specifically labeled “Attendant

Paid Jackpots” that accumulates the total value of credits paid by an attendant resulting from

a single winning alignment or combination, the amount of which is not capable of being paid

by the machine itself. This does not include progressive amounts or amounts awarded as a

result of an external bonusing system. This meter is only to include awards resulting from a

specifically identified amount listed in the manufacturer’s par sheet;

C. Slot Statistics Example.

The following example shows how these terms apply to real-world events.

♦ I insert a $50 bill into a $1 denomination slot machine

♦ The slot machine shows that I have a balance of 50 credits

♦ At this point, since I have not committed a bet:

o Coin In, Coin Out, Jackpots and Act Win are all $0

♦ I commit a $1 bet by pressing the PLAY button

♦ 1 credit = 1 dollar is deducted from my credit balance (50 credits  49 credits)

♦ I get a losing outcome (e.g. Blank-Seven-1Bar)

♦ At this point, my slot statistics are:

o Coin In = $1

o Coin Out = $0

o Jackpots = $0

-11-

o Act Win = $1 (i.e. the casino won a dollar and I lost a dollar)

♦ I commit another $1 bet by pressing the PLAY button

♦ 1 credit = 1 dollar is deducted from my credit balance (49 credits  48 credits)

♦ I get a winning outcome that pays 5x my bet (e.g. Cherry-Cherry-Blank)

♦ At this point, my primary slot statistics are:

o Coin In = $2

o Coin Out = $5

o Jackpots = $0

o Act Win = minus $3 (i.e. for this overall play session,

the casino is down $3 and I am up $3)

♦ Note that my machine-paid award went directly to my credit balance on the machine:

48 credits at the start of the proposition plus 5 credits award results in a new credits

balance of 48 credits + 5 credits = 53 credits

♦ If I press the CASH OUT button on the machine at this point, I would receive $53 in

some form (coins, token, vouchers, etc.) that I can carry away from the machine. The

process of cashing out would not change my primary slot statistics. Namely, I placed

$2 in wagers, received $5 in awards, and came out ahead $3 (which of course means a

corresponding $3 loss to the casino).

♦ Alternatively, I can continue playing on this same machine and can recirculate award

credits awarded by earlier wins into new wagers. For example, with 53 credits in my

balance, I have enough money to be able to play at least 53 more times, at $1 per

play, without having to insert any additional money.

♦ Let’s say, as quite often happens to players in real life, I continue playing until my

balance drops to 0 credits. Because I can, and do, recirculate winning credits back

into new wagers, my original $50 buy-in can result in my being able to play over 100

-12-

games and maybe over 200 games or more until my balance drops to 0. The more

wins I have, the more money I have to bet, the more games I can play without having

to insert any additional money into the machine.

♦ In the case where I was able to play 180 games until my balance drops to 0, my

primary slot statics would be:

o Coin In = $180

o Coin Out = $130

o Jackpots = $0

o Act Win = $50

In other words, my original $50 allowed me to play 180 games.

At $1 per game, this means I wagered $180. Of the $180 of my wagers, $50 was from my

original deposit and $130 was from awards paid to me from which I elected to make additional

bets.

$50 original buy-in amount + $130 recirculated awards = $180 total wagers.

At the end of this play session, I walk away from the machine with $50 less money in my

pocket than when I arrived (which likewise means the casino now has $50 more money). For this

example play session, my net gambling loss is therefore $50 and, likewise, the casino’s win is

$50.

Looking at the first column of data titled “2004” in Exhibit 2 we see the following data:

Coin In 8,517,492

Coin Out 7,034,435

Jackpots 1,114,645

Act Win 368,413

Acres Advantage Patron Management Report Specifics.

As discussed above, these data indicate that for

player-tracked machine

easily calculated:

Therefore,

’s net

2004 player-tracked gambling profit (loss) is

from staff-paid awards, or $7,034,435+$1,114,645 =

$8,149,080 of total awards.

$8,517,492 in wagers, resulting in $7,034,435 from

machine-paid awards and $1,114,645

play (i.e. gambling while using her Player Tracking

Card,)

made

and shows that

suffered a net gambling

loss

of $368,412 for the year 2004

-13-

while using her Player Tracking Card,)

made $2,846,048 in wagers,

These data indicate that for

player-tracked machine play (i.e. gambling

Act Win

129,691

Jackpots

499,040

Coin Out

2,217,317

Coin In

2,846,048

In similar fashion, the first column of data titled

“2005” in Exhibit 2 shows

the monetary values printed in the report.)

gambling win can be explained by rounding errors from not having the cents portion of

$368,41

gambling loss

from player-tracked play’s corresponding $368,41

$368,413 (Act Win) from her for 2004. (The $1 discrepancy between

’s

loss for 2004 correctly corresponds with the Casino reporting having won

from

Casino from player-tracked play.

’s $368,412 gambling

-14-

resulting in $2,217,317 from machine-paid awards and $499,040 from staff-paid awards,

or $2,217,317 + $499,040 = $2,716,357 of total awards.

(Act Win) from her for 2005.

E. Summary.

IV. DISCUSSION OF POTENTIAL WINNINGS FROM UNTRACKED SLOT PLAY.

A. Purpose.

In this section, I calculate the odds that the Petitioner could have attained a given amount

of untracked profit, that is, profits made solely from machine-paid awards whilst playing without

a Player Club Card. The analysis is limited to machine-paid awards, thereby excluding staff-paid

awards, since such staff-paid awards would be in conjunction with W-2G winnings, which are

already accounted for.

Therefore,

’s net

2005 gambling profit (loss) is calculated:

and shows that

suffered a net gambling

loss

of $129,691 for the year 2005

from

Casino from player-tracked play.

’s gambling loss for

2005 exactly corresponds with the

Casino

reporting having won $129,691

Based on the Acres System Patron Management Report

shown in Exhibit 2,

suffered net gambling losses of $368,412

and $129,691 from player-tracked gaming

machine play at

Casino

for the years 2004 and 2005, respectively.

-15-

B. Overall Approach.

I use an industry-standard confidence interval test to solve the stated problem. I obtain

precise inputs relating to slot machine payout behavior for this test by analyzing the data

available in Exhibit 2. I do not have to rely upon any assumptions or industry-based estimates

regarding game payout as I am able to directly measure a very large sample of play from the

Exhibit 2 data.

The value that I do not know, which is n, the number of untracked games played, I

examine as a variable, not a constant. Specifically, I obtain a result which indicates the required

probability for any number of games played without a Player Club Card. Furthermore, I

specifically a) identify the number of games most likely to produce the lowest test metric within

this scenario most likely to attain the required break-even profit and b) calculate the

corresponding probability. This covers the Petitioner’s worse-case scenario.

C. Analysis Approach.

The basic equation that applies in such a situation is the confidence interval equation:

(3)

z sd

∆ =

Where:

♦ ∆ or “delta” is the difference between actual and expected

♦ z is the normal distribution z-score which represents the number of standard

deviations from the mean our range is

♦ sd is the standard deviation

♦ n is the number of games played

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Equation (3) can be rearranged to calculate a z-score:

(4)

∆

A given z-score indicates the probability of natural occurrence of the measurement at

hand relative to the standard normal distribution equation:

(5)

probability of natural occurrence =

( ) 1

f z e

−

= −

A z-score above 3.0 is considered improbable. In most statistics text books, calculated z-

score tables usually only go as high as 3.0. A z-score of 6.0 is considered extremely

improbable, with odds in excess of 1 billion to 1 against natural occurrence.

To calculate the value for sd required for equation

(4)

, I follow this standard

definition equation for weighted standard deviation:

(6)

standard deviation =

variance

(7)

variance =

( )

( )*

p i x x

 

−

 

∑

Where:

♦

( )

p i

is proportion of occurrence of event i

♦

is the value of event i

♦

is the mean value of the distribution

I apply equations

(6)

and

(7)

to the report data from Exhibit 2, as shown in

Exhibit 3. The terms used for equation

(7)

♦

( )

p i

= Coin In for Month i / Total Coin In

♦

= Actual Base RTP for Month i

♦

= Total Theo Base RTP

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♦

RTP, or Return to Player, represents the average proportion of a

player’s wagers returned to player in winnings

actual machine-game records is 3.29%.

The ∆ value required to solve equation

(4)

needs to be calculated relative to the

enjoyed sufficient gambling profit from machine-only pays (i.e. awards not

paid by hand and recorded on a W-2G form) while not using her Player Club Card, to

compensate for the total net gaming losses indicated by the Patron Management report in

or more from machine-only pays when she was not using her Player Club Card?

To achieve a given level of total winnings from machine-only pays, I need to

calculate ∆, the difference between target wins and theoretically expected wins, as

follows:

(8)

∆ = | (RequiredRTP – ExpectedRTP) |

Where RTP (Return To Player) is defined as the ratio (Coin Out) / (Coin In).

The RTP required to generate $129K of profit from a given Coin In can be

expressed as:

(9)

RequiredRTP = (TargetProfit + Coin In) / Coin In = (TargetProfit /Coin In)+1

Therefore

(8)

can be expressed as:

(10)

∆ = | (TargetProfit / Coin In) + 100% - ExpectedRTP |

As shown in Exhibit 3,

sd, the standard deviation of RTP derived from

’s

question to be answered. Specifically, my objective is to calculate the odds that

Exhibit 2. Specifically, what are the odds that in

2005,

earned $129,691

-18-

By definition, expectedWin% = 100% - expectedRTP = the expected return to casino =

expected casino win. Specifically, expectedWin% = 100% - 82.1% = 17.9%. Additionally, the

value for total player wagers, Coin In, can be expressed as n * averageBet.

Therefore

(10)

can be further simplified as

(11)

∆ =

TargetProfit

ExpectedWin%

*AverageBet

Note that the absolute value function in

(8)

and

(10)

are no longer required in

(11)

since

all terms are positive numbers, which guarantees that the result will likewise be a positive

number.

D. Analysis Calculation.

In order to solve

(11)

, as well as to obtain the value n (the number of games) for

typical slot machine wager was between $10 and $20 per game, but to be conservative, I

assume an even larger average machine-game wager of $25. This is a conservative choice

because the larger the wager is, the easier it is to achieve a certain win threshold within a

certain number of plays.

Therefore, by substituting the following values:

♦

TargetProfit = $129,691

♦

AverageBet = $25

♦

Expected Hold = 100% - 82.1% = 17.9%

into equations

(11)

, I calculate ∆ as:

(12)

∆ =

$129,691

17.9%

*$25

equation

(4)

, I need to use a reasonable bet size.

informed me that her

-19-

Substituting this calculation for ∆ into equation

(4),

along with the knowledge that

sd = 3.29%, yields:

(13)

$129,691 5178.64

17.9% * 17.9% *

*$25

3.29% 3.29%

n n

   

+ +

   

   

= =

The absolute minimum number of games in which one could earn $129,691 or

more with awards no larger than $1,199 is

$129,691

109

$1,199

 

 

games.

-20-

By plotting equation

(13)

for n=109 through n=20,000, we get the following curve

(14)

It can be easily seen that the z-score value never drops below 17.5. Using differential calculus

the actual minimum z-score is 18.524 which occurs at n=28,981. Note that I am not implying the

Petitioner actually played these many games without her Player Club Card. What I am saying is

that the Petitioner would have had the maximal chance of enjoying profits from sub W-2G

winnings had she played these number of games without her Player Club Card.

To find a minima for z = f(n), we differentiate f(n) and solve f’(n) = 0. Specifically, we differentiate f(n)=

( )

1 1/2 1/2 1/2

1 5178.64

17.9% 30.39 5178.64 17.9% 157, 679 5.44

3.29%

n n n n n

− −

  

+ = + = +

  

  

to obtain f’(n)=

(

)

(

)

3/ 2 1/2

1 1

2 2

157,679* 5.44* 0

n n

− −

− + =

, or

( 157, 679 * 5.44) 0

− + =

, which has

two solutions: n=0 is one solution, the other is found by solving 5.44=157,679n, or n=5.44/157,679= 28,981.23.

-21-

As discussed earlier, a z-score of 6 is considered extremely improbable – on the order of

one chance in a billion. Normal distribution z-score tables typically published in statistics books

and mathematical handbooks usually only indicate the probability equivalent of z-score values

up to 3.0 (also known as “3 sigma”). One statistics book does list the odds for z=5 to be in excess

of 1 in 3 million. The NORMSDIST(z) function in the Microsoft Excel 2007 spreadsheet

program, does not work for an input z-score value of 7.88 or higher. At z=7.87, the Microsoft

Excel 2007 NORMSDIST(z) function generates the value 562,949,953,421,312, which, in other

words, indicates odds in excess of 1 in 500 trillion. An outcome with a z-score of z=18.5 is so

astronomically improbable as to be safely considered impossible that the Petitioner could have

earned $129K in machine-paid profits from any amount of non-tracked slot gambling (i.e. from

profits made solely from machine-paid awards whilst playing without a Player Club Card).

Obviously, if the odds of earning $129K in these circumstances are astronomical, then the

odds of earning an even larger amount, $368K in 2004 (which is even more unlikely), are

likewise also astronomical.

-22-

E. Analysis Summary.

The analysis demonstrates the impossibility of the Petitioner attaining machine-paid gambling

winning (i.e. where all awards are below the W-2G reporting threshold) on the order of $129K or

larger when she did not use her Player Club Card, irrespective of the number of games she

played without her Player Club Card.

Respectfully Submitted,

______________________________________

MARK C. NICELY Dated:

EXHIBIT 3

DATA AND CHART USED FOR CALCULATION OF BASE GAME

STANDARD DEVIATION –