Snake Eyes
True Odds versus Vegas Payout
Hey wonderful people
Hope you had a great 2024, Happy New Years to you and your wonderful family. I started writing ADCB last December and it has been such a pleasure sharing my love for quant and finance with you. I deeply appreciate you reading this space, I’ll always treat your time with the respect that it deserves and if you know of anyone who’d love this space, please share my stack with them
As I said a couple of days ago, I’m touching base with you from Las Vegas
I mostly sleep for nine to ten hours a night, walk the strip in the mornings, and people-watch in different casinos for the rest of the day. I love to watch stand-up comedy and see the inner-workings of Caesars Sportsbook in real time, so I did those too
I also grew a holiday beard which I thought made me look more wiser to the devious ways of the world, as I ponder about my life with Cognac and cigars on the night streets of Vegas. I feel home here … I feel a connection here
I have always wanted to understand the pulse of Sin City, and I’m fascinated by the psychology of gambling and how people think about chance and probabilities. So, in a lot of ways, for what I’m interested in - this is homecoming
I walked all the way to one end of the strip to - The Welcome to the Las Vegas Sign yesterday, and it did not disappoint. There is a tender and nostalgic ol’ school Americana to it, especially as the sign glistens under the evening sun just before sunset. If anything, it gets forever-more beautiful after sundown - when the blinking night-lights around the sign adorn its structure
‘tis not ol’ school like crusty, dusty and outdated, but ol’ school like vintage, classic and timeless - like Nat King Cole or Bob Dylan, Rita Hayworth or Marilyn Monroe
Only fitting, to the sign’s very right, there is a giant Harley Davidson showroom. Is there anything more fitting than seeing a majestic Harley store just as we are about to leave the Las Vegas strip, and venture into the backroads of the desert south-west ? I could almost smell the burnt tires on the pavement, those black leather jackets and long stringy-blonde locks emanating out of those beaten-up and faded black helmets
… add oversized sunglasses on sunburnt faces with handlebar mustaches - for good measure as well.
Hey - it’s not my fault, this is what decades of watching Hollywood’s finest taught me, I’m hard-wired to think about biker dudes and dudess this way. Sons of Anarchy, The Lincoln Lawyer … you get the drift
It reminds me of everythin’ that is wonderful about ol’ school Americana … and everythin’ that is barely legal
Casinos are a lot of things, the gamut runs from delightful to depraved - but they also lend themselves for us to think about chance and odds … if we only paid attention to what is imprinted in white bold letters on those gorgeous dark-green felt - that is a casino table
If ever, you want someone to understand probability and statistics in real time, and want them to think about if the return on the risk they undertook is rewarded in commensurate measure, bring them to the floor of Caesars Palace and teach them real time … you can take them to the Bellagio as well
Lets use the Craps table in Caesars Palace for instance - you don’t need to know craps, it’s sinfully easy, I’ll teach you
Craps is betting on the outcome of a single roll of two (hopefully) six sided fair dice - that’s about it. If you bet its going to be 1 and 1, and if you do get 1 and 1 - you win
This is an actual Craps table from Caesars Palace - and you can read the odds for the Horn Bet
We don’t really need to get into the mechanics of Craps for us to understand this. All you need to know is, if you place $ 4 in a horn bet
Then you are betting
$ 1 on snake eyes (we call 1 and 1 as snake eyes)
$ 1 on getting 6 and 6
$ 1 on getting 1 and 2
and $ 1 on getting 6 and 5
- in the next single roll of the two, six sided fair dice
Do you remember when they rig the dice in Oceans 13, so it gets them the two numbers they want ? this is why they do it
If you get snake eyes, your $ 1, will essentially give you a payout of 31x (including your initial stake of $ 1, which means your profit is essentially $ 30)
Perfect
From this point, we’d only focus on snake eyes to understand if the payout is fair, for the risk you undertook in taking on a horn bet
Great
In a six sided fair dice
the chance of you getting 6 in a single roll is 1/6
the chance of you getting a second 6 in the other dice is also 1/6
Since the outcomes of both dice are independent of each other
So, P(Winning, for getting snake eyes) = 1/36
For the odds to be fair, we typically think of Expected Value to be 0. As in, when you play this game many times, you will neither gain nor lose, and hence the odds are fair. So, we just solve this by setting E(x) = 0
Think of your total payout to be C. If you get snake eyes, you’ll get a profit of C-1, and if you lose, you’d simply lose the $ 1 you bet on getting snake eyes. You did bet $ 4, but we are only considering how much you implicitly bet on snake eyes here, which is $ 1, yeah ?
When you solve for C, you’ll know what your actual payout should be (based on true odds, as opposed to what Vegas gives you)
True Odds versus House Payout
So, the true odds of snake eyes is 1/36, which means your payout (including the initial stake of $ 1) should be $ 36 - if the payout is fair
But as you can see from the table, Caesars Palace only offers you $ 31 in total for snake eyes
We call this the house edge, and the house does this to essentially maintain its long term profitability. The house almost always, gives you less than what you deserve, for the level of risk you did take on - in placing a bet. Snake eyes is also a terrible bet, a 1 in 36 chance ? why don’t you just burn your dollars and go home ?
In essence, even when you win and beat really long odds, the house still wins because they pay you less than what you deserve, for the magnitude of risk you undertook
But you didn’t come to Caesars Palace with a distorted notion that this is a fair place did you ?
The pit boss has multiple boyfriends (or girlfriends) and those Christmas and New Years gifts aren’t going to send themselves
He got bills to pay chief, he got tons of bills to pay … and if you bet on snake eyes, you are paying it for him
Welcome to Las Vegas !