By Matt Kaufman

Countless articles have recently circulated surrounding the gambling habits of Stephen Paddock, the mass murderer behind the recent Las Vegas shooting in which 58 lives were lost and hundreds of others were injured.

The most prominent example of media speculation on Paddock’s gambling habit was likely NBC News, which on Friday ran a story containing the following information:

“IRS records show that Paddock was a successful gambler, earning at least $5 million in 2015.”

Critics of NBC’s reporting were quick to point out that the IRS figure is more likely than not to contain earnings, as opposed to his net winnings on gambling. Most people then conclude that it would be impossible to know how much Paddock won or lost because we don’t have enough information.

However, if we’re willing to make some reasonable assumptions, we can actually deduce quite a bit more than that using nothing more than publicly available information and math.

How gambling earnings are reported in Las Vegas

In Las Vegas, casino game earnings only generate W-2G tax forms when a player receives a single payout of a specified amount which varies depending on the game being played. For video poker, that number is $1,200.

That means a tax form is only generated when a player hits a payout of $1,200 on a single hand of video poker (excluding the value of the wager itself). A W-2G is not triggered when cashing out any amount from a machine - hypothetically you could cash out from a machine having won $100,000 (or any other number over $1,200) without generating a tax form if you managed to do it without winning $1,200 on any individual hand.

The IRS is notified by casinos of all W-2Gs for each calendar year. Throughout this article, I will be operating under the assumption that Paddock’s $5 million figure represents the total value of W-2Gs that casinos issued to him in 2015, as that seems far-and-away to be the likeliest explanation of the figure.

What else can we derive from facts about Paddock’s gambling?

Be warned: you will need to understand a basic definition of “expected value” to be able to understand some of the rest of this. Skip this paragraph if you already get it. Perhaps the easiest way to think about expected value in a gambling context is that it’s the average amount that would be gained or lost over time if you were to repeat an event over and over infinitely. If a video poker machine has a 99% return, the expected value you’ll receive when playing a $1 hand on that machine is 99 cents because that’s the average you’ll get back for your dollar - even though you can’t actually receive exactly 99 cents on any of the individual hands.

Let’s outline several details of Paddock’s gambling from widespread reporting:

He played at the $25 denomination He played at Mandalay Bay He hit $5 million in W-2G generating events

At the $25 denomination, he almost certainly was playing max credits (5) for a total of $125/hand. Many outlets have reported he played for “$100-a-hand,” but anyone who knows anything about video poker would tell you that no advanced gambler would play for less-than-max credits.

Next, we need to figure out how much he really earned if $125/hand resulted in $5 million in W-2Gs.

To do that, let’s first look at the paytable for the machine he played at Mandalay Bay. The best return on video poker available at the property, at the denomination he played, is 8/5 Bonus Poker with a 99.17% return (so the expected return a player gets back each hand of play is 99.17% of what they had wagered).

Playing max credits, 8/5 Bonus Poker returning 99.17% would look like this:

Hand Return in credits per credit wagered Real money received when playing $125/hand Return % that corresponds to making this hand Royal Flush 800 $100,000 0.019884 Straight Flush 50 $6,250 0.005342 4 Aces 80 $10,000 0.015667 4 Deuces through Fours 40 $5,000 0.021090 4 Fives through Kings 25 $3,125 0.040997 Full House 8 $1,000 0.092114 Flush 5 $625 0.054400 Straight 4 $500 0.044885 Three-of-a-kind 3 $375 0.223405 Two Pair 2 $250 0.258616 Jacks or better 1 $125 0.215259 Nothing 0 $0 0.000000

The return percentages, taken from Wizard of Odds, is the part of the total 99.17% return that corresponds to making each individual hand ranking.

So, why does the paytable matter? Because we can see that tax forms would only generate when Paddock made four-of-a-kind or better - full houses only pay $1,000 at this denomination.

Next, let’s add up the returns for all of the hands that generate a tax form. The returns for making four-of-a-kind or better add up to 0.10298 - meaning those events happening account for just over 10% of the return on this machine.

It also means that they account for (0.10298/0.99166) 10.384608% of the total expected return of this machine.

Note: the total return for the machine was previously mentioned as 99.17%, but is more accurately 99.166%, so I’m using the precise number above, and will use it again later.

That means the $5 million in earnings that was reported to the IRS is likely quite a lot larger than that. The $5 million reported is 10.384608% of the amount he probably really earned.

Here’s the first big conclusion: If the IRS believes Stephen Paddock earned $5 million in 2015 playing video poker, his gross earnings were probably more like $48,148,182.

Of course, that’s excluding all the hands he lost.

So how much did he lose?

On a video poker machine with an expected return of 99.17%, a player only receives that full 99.17% return if they make zero mistakes. Even advanced video poker players make an occasional button-pressing mistake on account of the speed with which they play, but for the sake of argument let’s assume from this point that Stephen Paddock was an excellent video poker player who never made a mistake.

If $48,148,182 was Paddock’s expected earnings on a 99.17% machine, the expected value of his coin-in was $48,553,115. He probably lost the difference between those figures, minus the value he earned from being a VIP player and receiving comped vacations and possibly cash back in other forms.

The $5 million in earnings that NBC reported is quite unlikely to be an exact figure, and therefore neither is this conclusion. But here goes nothing: if the IRS believed Stephen Paddock to have earned $5 million playing video poker in 2015, his real-world likely loss was probably about $404,933. But he got lots of hotel rooms, show tickets, meals, and some amount of cash back from the casino.

We can still figure out more: his expected loss per hand was $1.0425 ((1-0.99166)*125), and he therefore probably played about (404,933/1.0425) 388,425 hands.

We’ve read he played at a very fast pace, but not knowing exactly how fast we can assume he played at a quick pace for a serious player. Video poker forums around the internet suggest 800 hands per hour is a reasonable guess on pace. Using that as a best guess, he probably played for about 485 and a half hours - just over 9 hours per week for that year, if it was evenly distributed. Although it’s easier to imagine 6 separate week-long vacations during which he spent 12 hours a day on the machines. But of course, by now I’m purely speculating.

In fact, all of the above is speculation. But my response to all of the debate on Stephen Paddock’s gambling habits is this: let’s not remember him as some mastermind gambling winner - it’s a much better guess that he wasn’t one.

Matt Kaufman is a Las Vegas-based gaming consultant and the editor of numerous gambling websites including https://bonuscodepoker.com, https://bonusreferrercode.com, and http://www.parttimepoker.com/.