COMMON CASINO GAMES: MEAN, VARIANCE AND STANDARD DEVIATION
COMMON CASINO GAMES: MEAN, VARIANCE AND STANDARD DEVIATION

 

Figure 9 gives a list of many well known table casino games, their mean result for a single round wagering 1 unit, their standard deviation and variance.

 

Single Round Wager

The wagers in Figure 9 with the highest volatility are, in this order, Mississippi Stud, the Dragon 7 side bet in EZ Baccarat, playing a number straight-up in roulette, Let it Ride, and a 2 or 12 proposition bet in craps. There is a lot of similarity to playing a number straight-up in roulette (odds are 1-in-38 and pays 35-to-1) and playing a proposition bet on 2 or 12 in craps (odds are 1-in-36 and pays 30-to-1). Hence these two wagers have very similar values for their standard deviation. High variance games and wagers are for those who like a lot of volatility. The results are going to be all over the place in a session. Wild things are likely to happen. There will be big jackpot hands and long losing streaks. There won’t be much, if any, back and forth, winning and losing one bet at a time. The high value for the variance in Let it Ride has been problematic for the game. It is almost as wild a ride as playing a number straight-up in roulette. Moreover, a player cannot choose any method of playing the game that reduces his variance. Let it Ride has dropped from a high of 221 tables in Nevada in 1996 to 121 tables in 2009 (see Appendix C). Let it Ride continues to lose market share because of its unshakeable volatility. Players lose too quickly while hoping for that jackpot hand.

The game Mississippi Stud has a pay profile similar to Let it Ride; I’ve heard it referred to as “Let it Ride in reverse.” However, Mississippi Stud also has nearly twice the standard deviation as Let it Ride, making it by far the most volatile non-traditional table game in wide distribution. Mississippi Stud is demographic specific. It works best in casinos with a lot of different carnival games; players can drift to it when they want a certain experience, but few sit at it for a long time. It quickly drains players who must play 4 wagering units per round to chase a possible win. Any time a player is dealt a high pair, he instantly wins 10 units, possibly a lot more. The single highest volatility hands for this game are four-ofa-kind and a royal flush. The pay table is highly skewed towards the top end; players lose a lot while pursuing high paying rare hands. It is impossible for a player to win less than four bets at a time; losing streaks can be long and tortuous.

CASINO GAMES: MEAN ( ), STD. DEV.( ), VARIANCE ( )

The games with the lowest volatility are, in this order, Pai Gow Tiles, Pai Gow Poker, baccarat (Banker), EZ Baccarat (Banker), baccarat (Player), roulette (even/odd) and craps (don’t pass/don’t come). These games are for those who shun variance and want the lowest volatility. Players’ results will move slowly up and down. These games will be safe: players don’t have to worry that big events are going to suddenly change their session results.

Dragon

At the extreme of volatility for blackjack side bets is the Lucky Ladies side bet (not included in the table). For a six deck game with pay table 4-9-19-125-1000, the Lucky Ladies side bet has µ = 0.7529 and σ = 4.96. It has a very high house edge of 100 × (1 – 0.7529)% = 24.71% (the Nevada state legal maximum house edge is 25%). I am not aware of any table game or side bet (or slot machine, or keno pay table or video poker game) that has a higher house edge.

A side bet that is even more volatile than Lucky Ladies is the Dragon 7 side bet in EZ Baccarat. A player wins this bet by making a wager that a Banker three-card total of seven will beat the player’s hand. The odds are about 1-in 44.38 and this side bet pays 40-to-1. The Dragon side bet has µ = 0.9239 (house edge of 7.61%) and σ = 6.09.

Most other side bets, no matter the game, are much less volatile than Lucky Ladies in blackjack or the Dragon side bet in EZ Pai Gow. In general, only progressive and streak side bets have a higher level of volatility. Take note of the standard deviation for the roulette wagers listed in Figure 9. These illustrate the wide range of variance the player can choose when playing roulette. The player can wager on even/odd, a column, a number straight-up, or he can mix and match as he decides. The house edge never changes.

Baccarat BetsThe four wagers listed for craps in Figure 9 illustrate that a player can pick and choose his variance. However, in craps, in order to get high variance without taking “odds,” the player will have to play high house edge wagers. This is why odds are so popular in craps; odds are pure variance, nothing more. The house edge for all odds wagers in craps is 0.00%, the theoretical RTP is 100%. Over the long run, the player expects to break even on his odds wagers. Most casinos offer odds; some use it as a marketing tool, allowing up to 100 × times odds to be purchased. The primary purpose of offering odds is to allow the player to have the experience of volatility he wants at no additional cost to the player or casino.

 

 

By
Eliot Jacobson Ph.D.