Remembering The Riviera

On May 4, a 60-year run on the Las Vegas Strip will end when the Riviera Hotel and Casino closes, soon to be demolished to make way for an expansion of the Las Vegas Convention Center. The Riviera’s place in Las Vegas history is secure: it was the first high-rise hotel on the Strip, was part-owned by Dean Martin, played host to Liberace, and (in this writer’s opinion) was home to the best comedy club in Las Vegas, among other noteworthy events.

In recent years, there was an interest in offering good gambling at the Riviera, which included some of the best craps odds available. Dice aficionados know that the “free odds” wager backing up a Pass or Don’t Pass bet is the best bet ordinarily available in a casino, as it’s paid off at true odds with no house advantage. While this is a great deal for gamblers, its impact is usually tempered somewhat by a restriction on the amount of money that may be risked in an odds bet. This is often stated in a form such as “3X/4X/5X odds”, which means that you may risk three times your initial wager on an odds bet if the point is 4 or 10, four times that wager if the point is 5 or 9, and five times your original bet if the point is 6 or 8.

Consider an odds bet with a point of 4. Once the point 4 has been established, there are three ways to roll a 4 and six ways to roll a 7; all other rolls fail to resolve the main Pass line bet. Accordingly, the probability of winning this bet is 3/9, and the odds against winning are 2-1. The odds portion of this wager pays off at this same 2-1, making the house edge 0% on that part of the bet alone. The corresponding payoffs for a point of 5 or 9 are 3-2, and odds bets backing a point of 6 or 8 pay off at 6-5. (Fractions are typically rounded down, in favor of the casino; if you cannot remember the different payoffs, a workable strategy is always to make odds bets in multiples of $10.)

At this multiplier level, odds bets are one of the best wagers you can find in a casino. In 2012, the Riviera made odds much more lucrative by launching a craps promotion offering 1000X odds, regardless of the point. If you put out $5 on the Pass line and a point is established, you could back that up with up to $5000 in odds. While your initial bet would pay off at 1-1 if the point was made, the odds portion of your wager would be paid off according to the odds above, which has the effect of driving down the house advantage on the total bet from the already-low 1.41% applying to a Pass line wager.

A Pass line bet with odds, with any multiplier, does not change the expected return or the odds of winning; what does change is the house edge relative to the amount wagered. Since more money is at risk, the house advantage decreases. In the case of a $5 odds bet with 1000X odds, the expected return incorporating the comeout roll and all possible outcomes is -$7/99, or about -7.07 cents. Dividing this by $5 gives the house edge of 1.41% that we expect from a Pass wager. Dividing by $5005, assuming a full odds wager, shows that the new house edge is .0014%–which is as close to an even game as you’re likely to find at the craps table.

While odds bets offer a legitimate way to decrease the casino’s advantage, there is, of course, always a catch in an opportunity like this. Because the point has been established when the time comes to place the odds part of the wager, you are at a disadvantage when your odds chips hit the felt. While the idea of risking $5005 at such a tiny disadvantage may appear attractive, the fact is that you’re leaving yourself open to some wild bankroll fluctuations when you make a bet this large. If the point is 5 or 9, for example, you’re either going to win $7505 or lose $5005 on every decision–and most gamblers aren’t going to be able to withstand the inevitable losing streaks that arise at the dice table.

In short, it’s not the average loss of $1 per bet that gets you, it’s the standard deviation (the amount that a “typical” decision lies from the mean) of $6128.62.

Arrivederci, Riviera. You may not have had the bells and whistles of later Strip megaresorts, but you offered a good gamble, and were fun while you lasted.

Read more about the mathematics of gambling in Mark Bollman’s Basic Gambling Mathematics: The Numbers Behind The Neon, available from CRC Press or at

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