• Appendices
• Craps Analysis
• Miscellaneous

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Introduction

Live Poker Night is the first of many planned social games under the Live Game Night banner, with FlowPlay planning to launch Bunco and Mahjong over the next 90 days. They also have an extensive live casino section. Although Grand Poker hasn't become terribly popular, it should still be considered a secondary option for US-based online poker players. You can find active cash games up to the $1/2 NL level and, as tends to be the case. They also have an extensive live casino section. Although Grand Poker hasn't become terribly popular, it should still be considered a secondary option for US-based online poker players. You can find active cash games up to the $1/2 NL level and, as tends to be the case with sports and casino sites, the players are quite soft.

Introduction

Welcome to the craps appendix. This is where I derive the player's edge for all the major bets in craps. Outside of this appendix I usually speak about the house edge, which is just the product of the player's edge and -1. To avoid multiplying by -1 for every bet I shall speak of everything in term's of the player's edge, which you can expect to be negative since the house ultimately has the edge on all bets except the free odds. Please stay a while and work through some of the bets yourself. Not only will this give you a deeper understanding of the odds but hopefully motivate you to refresh or improve your math skills.

Before going on you must have an understanding of the probability of throwing each total in one roll. This is explained in depth in my dice probability basics page. If you didn't know or can't figure out that the probability rolling a 6 is 5/36 then a visit to that page is a prerequisite for this page.

The general formula for the expected return of a bet is:

∑ (probability of event i) × (return of event i) over all possible outcomes.

The player's edge is the expected return divided by the initial bet. For example when betting against the line on a sporting event you have to bet $11 to win $10. Assuming a 50% chance of winning the expected return would be 0.5×(10) + 0.5×(-11) = -0.5 . The player's edge would be -0.5/11 = -1/22 ≈ -4.545%.

An exception to the house edge rule is when a tie is possible. In general ties are ignored in house edge calculations. To adjust for this, when a tie is possible, divide the expected return by the average bet resolved. The 'average bet resolved' is the product of the initial wager and the probability that the bet was resolved. In craps the only bets with a tie are the don't pass and the don't come.

Many of the bets in craps win if one particular event happens before another. These bets can take several rolls or more to resolve. If a wager wins with probability p, loses with probability q, and stays active with probability 1-p-q then the probability of winning eventually is:

∑ p×(1-p-q)ⁱ (for i=0 to infinity) =
p × (1/(1-(1-p-q))) = p × (1/(p+q)) = p/(p+q).

Throughout this page you will see a lot of expressions of the form p/(p+q). To save space I do not derive the expression each time since it is worked out above.

Pass/Come

The probability of winning on the come out roll is pr(7)+pr(11) = 6/36 + 2/36 = 8/36.

The probability of establishing a point and then winning is pr(4)×pr(4 before 7) + pr(5)×pr(5 before 7) + pr(6)×pr(6 before 7) + pr(8)×pr(8 before 7) + pr(9)×pr(9 before 7) + pr(10)×pr(10 before 7) =

(3/36)×(3/9) + (4/36)×(4/10) + (5/36)×(5/11) + (5/36)×(5/11) + (4/36)×(4/10) + (3/36)×(3/9) =
(2/36) × (9/9 + 16/10 + 25/11) =
(2/36) × (990/990 + 1584/990 + 2250/990) =
(2/36) × (4824/990) = 9648/35640
The overall probability of winning is 8/36 + 9648/35640 = 17568/35640 = 244/495
The probability of losing is obviously 1-(244/495) = 251/495
The player's edge is thus (244/495)×(+1) + (251/495)×(-1) = -7/495 ≈ -1.414%.

Don't Pass/Don't Come

The probability of winning on the come out roll is pr(2)+pr(3) = 1/36 + 2/36 = 3/36.
The probability of pushing on the come out roll is pr(12) = 1/36.
The probability of establishing a point and then winning is pr(4)×pr(7 before 4) + pr(5)×pr(7 before 5) + pr(6)×pr(7 before 6) + pr(8)×pr(7 before 8) + pr(9)×pr(7 before 9) + pr(10)×pr(7 before 10) =
(3/36)×(6/9) + (4/36)×(6/10) + (5/36)×(6/11) + (5/36)×(6/11) + (4/36)×(6/10) + (3/36)×(6/9) =
(2/36) × (18/9 + 24/10 + 30/11) =
(2/36) × (1980/990 + 2376/990 + 2700/990) =
(2/36) × (7056/990) = 14112/35640
The total probability of winning is 3/36 + 14112/35640 = 17082/35640 = 2847/5940
The probability of losing is 1-(2847/5940 + 1/36) = 1-(3012/5940) = 2928/5940
The expected return is 2847/5940×(+1) + 2928/5940×(-1) = -81/5940 = -3/220 ≈ 1.364%

Most other sources on craps will claim that the house edge on the don't pass bet is 1.403%. The source of the discrepancy lies is whether or not to count ties. I prefer to count ties as money bet and others do not. I'm not saying that one side is right or wrong, just that I prefer counting them. If you don't count ties as money bet then you should divide by figure above by the probability that the bet will be resolved in a win or loss (35/36). So 1.364%/(35/36) ≈ -1.403%. This is the house edge assuming that the player never rolls a 12 on the come out roll.

Place Bets to Win

Place bet on 6 or 8: [(5/11)×7 + (6/11)×(-6)]/6 = (-1/11)/6 = -1/66 ≈ -1.515%
Place bet on 5 or 9: [(4/10)×7 + (6/10)×(-5)]/5 = (-2/10)/5 = -1/25 = -4.000%
Place bet on 4 or 10: [(3/9)×9 + (6/9)×(-5)]/5 = (-3/9)/5 = -1/15 ≈ -6.667%

Place Bets to Lose

Place bet to lose on 6 or 8: [(6/11)×4 + (5/11)×(-5)]/5 = (-1/11)/5 = -1/55 ≈ -1.818%
Place bet to lose on 5 or 9: [(6/10)×5 + (4/10)×(-8)]/8 = (-2/10)/8 = -1/40 = -2.500%
Place bet to lose on 4 or 10: [(6/9)×5 + (3/9)×(-11)]/11 = (-3/9)/11 = -1/33 ≈ -3.030%

Note: These bets are not allowed in land casinos. They can only be found in some Internet casinos.

Buy

Buy bet on 6 or 8: [(5/11)×23 + (6/11)×(-21)]/21 = (-11/11)/21 = -1/21 ≈ -4.762%
Buy bet on 5 or 9: [(4/10)×29 + (6/10)×(-21)]/21 = (-10/10)/21 = -1/21 = -4.762%
Buy bet on 4 or 10: [(3/9)×39 + (6/9)×(-21)]/21 = (-9/9)/21 = -1/21 ≈ -4.762%

Lay

Lay bet to lose on 6 or 8: [(6/11)×19 + (5/11)×(-25)]/25 = (-11/11)/25 = -1/25 ≈ -4.000%
Lay bet to lose on 5 or 9: [(6/10)×19 + (4/10)×(-31)]/31 = (-10/10)/31 = -1/31 = -3.226%
Lay bet to lose on 4 or 10: [(6/9)×19 + (3/9)×(-41)]/41 = (-9/9)/41 = -1/41 ≈ -2.439%

Big 6/Big 8

[(5/11)×1 + (6/11)×(-1)]/1 = -1/11 ≈ 9.091%

Hard 4/Hard 10

Note: The hard 4 and hard 10 pay 7 to 1, or 8for 1. In craps the odds on the cloth are listed on a for 1 basis, including the graphic above.

The probability of a hard 4 on any given roll is 1/36.
The probability of a 7 on any given roll is 6/36.
The probability of a soft 4 on any given roll is 2/36 (1+3 and 3+1).
The probability of winning on any given roll is 1/36.
The probability of losing on any given roll is 6/36 + 2/36 = 8/36.
The probability of winning the bet is p/(p+q) (see above) = (1/36)/(9/36) = 1/9
The expected return is (1/9)×7 + (8/9)×(-1) = -1/9 ≈ 11.111%.
The player's edge is also -1/9 since the bet is 1 unit.
The odds are the same for a hard 10.

Hard 6/Hard 8

Note: The hard 4 and hard 10 pay 9 to 1, or 10for 1. In craps the odds on the cloth are listed on a for 1 basis, including the graphic above.

The probability of a hard 6 on any given roll is 1/36.
The probability of a 7 on any given roll is 6/36.
The probability of a soft 6 on any given roll is 4/36 (1+5, 2+3, 3+2, and 5+1).
The probability of winning on any given roll is 1/36.
The probability of losing on any given roll is 6/36 + 4/36 = 10/36.
The probability of winning the bet is p/(p+q) (see above) = (1/36)/(11/36) = 1/11
The expected return is (1/11)×9 + (10/11)×(-1) = -1/11 ≈ 9.091%.
The player's edge is also -1/11 since the bet is 1 unit.
The odds are the same for a hard 8.

Craps 2/Craps 12

[(1/36)×30 + (35/36)×(-1)]/1 = -5/36 ≈ -13.889%

Craps 3/Craps 11

[(2/36)×15 + (34/36)×(-1)]/1 = -4/36 ≈ -11.111%

Any Craps

[(4/36)×7 + (32/36)×(-1)]/1 = -4/36 ≈ -11.111%

Any 7

[(6/36)×4 + (30/36)×(-1)]/1 = -6/36 ≈ -16.667%

Horn

The probability of rolling either a 2 or 12 is 1/36 + 1/36 = 2/36.
The probability of rolling either a 3 or 11 is 2/36 + 2/36 = 4/36.
The probability of roling anything else is 1-2/36-4/36 = 30/36.
Remember that the horn bet is like all four craps bets in one. Even if one wins the other three still lose. The house edge is:
[(2/36)×27 + (4/36)×12 + (30/36)×(-4)]/4 = (-18/36)/4 = 12.500%

Field

When the 12 pays 2:1 the expected return is:
2×(pr(2)+pr(12)) + 1×(pr(3)+pr(4)+pr(5)+pr(10)+pr(11)) + -1×(pr(6)+pr(7)+pr(8)+pr(9)) =
2×(1/36 + 1/36) + 1×(2/36 + 3/36+ 4/36 + 3/36 + 2/36) + -1×(5/36 + 6/36 + 5/36+ 4/36) =
2×(2/36) + 1×(14/36) + -1×(20/36) = -2/36 = -1/18 ≈ 5.556%.

When the 12 pays 3:1 the expected return is:
3×pr(2) + 2×pr(12)) + 1×(pr(3)+pr(4)+pr(5)+pr(10)+pr(11)) + -1×(pr(6)+pr(7)+pr(8)+pr(9)) =
3×(1/36) + 2×(1/36) + 1×(2/36 + 3/36+ 4/36 + 3/36 + 2/36) + -1×(5/36 + 6/36 + 5/36+ 4/36) =
3×(1/36) + 2×(1/36) + 1×(14/36) + -1×(20/36) = -1/36 ≈ 2.778%.

Buying Odds

4 and 10: [(3/9)×2 + (6/9)×(-1)]/1 = 0.000%
5 and 9: [(4/10)×3 + (6/10)×(-2)]/2 = 0.000%
6 and 8: [(5/11)×6 + (6/11)×(-5)]/5 = 0.000%

Laying Odds

4 and 10: [(6/9)×1 + (3/9)×(-2)]/1 = 0.000%
5 and 9: [(6/10)×2 + (4/10)×(-3)]/2 = 0.000%
6 and 8: [(6/11)×5 + (5/11)×(-6)]/5 = 0.000%

Combined Pass and Buying Odds

The player edge on the combined pass and buying odds is the average player gain divided by the average player bet. The gain on the pass line is always -7/495 and the gain on the odds is always 0. The expected bet depends on what multiple of odds you are allowed. Lets assume full double odds, or that the pass line bet is $2, the odds bet on a 4, 5, 9, and 10 is $4, and the odds on a 6 or 8 is $5.

The average gain is -2×(7/495) = -14/495.

The average bet is 2 + (3/36)×4 + (4/36)×4 + (5/36)×5 + (5/36)×5 + (4/36)×4 + (3/36)×4] =
2 + 106/36 = 178/36

The player edge is (-14/495)/(178/36) = -0.572%.

The general formula if you can take x times odds on the 6 and 8, y times on the 5 and 9, and z times on the 4 and 10 is (-7 / 495) / [ 1 + ((5x + 4y + 3z) / 18) ]

Combined Don't Pass and Laying Odds

The player edge on the combined don't pass and laying odds is the average player gain divided by the average player bet. The gain on the don't pass is always -3/220 and the gain on the odds is always 0. The expected bet depends on what multiple of odds you are allowed. Lets assume double odds and a don't pass bet of $10. Then the player can lay odds of $40 for a win of $20 on the 4 and 10, $30 for a win of $20 on the 5 and 9, and $24 on the 6 and 8 for a win of $20. The average gain is -10×(3/220) = -30/220.

The average bet is 10 + 2×[(3/36)×40 + (4/36)×30 + (5/36)×24] = 30.

The player edge is (-30/220)/30 = -0.455%.

The general formula if you can buy x times odds then the house edge on the combined don't pass and laying odds is (3/220)/(1+x).

Net Gain/Loss per Session

The chart below shows the net gain or loss you can expect over 100 trials, or come out rolls. For purposes of creating the chart the player would bet $1 on the pass line and take full double odds.

Here are some actual numbers that show the probability of falling into various intervals.

Session Win/Loss

Interval	Probability
loss of over $100	0.0422%
loss of $76-$100	0.6499%
loss of $51-$75	4.6414%
loss of $26-$50	16.3560%
loss of $1-$25	30.0583%
break even	0.6743%
win of $1-$25	28.6368%
win of $26-$50	14.4257%
win of $51-$75	3.9097%
win of $76-$100	0.5639%
win of over $100	0.0418%

The graph and table were created by simulating 1,000,000 sessions of 100 trials, or come out rolls, and tabulating the results of each session.

Internal Links

• How the house edge for each bet is derived, in brief.
• The house edge of all the major bets on both a per-bet made and per-roll basis
• Dice Control Experiments. The results of two experiments on skillful dice throwing.
• Dice Control Advantage. The player advantage, assuming he can influence the dice.
• Craps variants. Alternative rules and bets such as the Fire Bet, Crapless Craps, and Card Craps.
• California craps. How craps is played in California using playing cards.
• Play Craps. Craps game using cards at the Viejas casino in San Diego.
• Number of Rolls Table. Probability of a shooter lasting 1 to 200 rolls before a seven-out.
• Ask the Wizard. See craps questions I've answered about:
  
• Simple Craps game. My simple Java craps game.

Written by: Michael Shackleford

Pr Games Live Poker Games

SEATTLE, May 19, 2020 /PRNewswire/ -- FlowPlay, creator of one of the most powerful immersive gaming platforms, today announced the launch of Live Game Night, a new online game providing one-of-a-kind, invite-only social play with live video chat. The first title in the Live Game Night collection, Live Poker Night, marks the return of poker night, allowing up to five of a host’s friends and family to play together in a face-to-face virtual matchup.

As more than 65 percent of Americans continue to live life under lockdown, and many are seeking new ways of connecting with others while at home, interest in online communities and social games is spiking. FlowPlay’s flagship social casino, Vegas World, recorded a 25 percent increase in players over the last 90 days. Live Game Night leverages state-of-the-art technology to provide as close to an in-person connection as possible - without the awkward small talk at your in-law’s. Initially launching on desktop, Live Game Night will soon be also available to play on mobile devices.

“Online games are so much more than entertainment. They are an escape from today’s uncertain times, a means for keeping in touch with old friends and a place to create new ones,” said Derrick Morton, CEO, FlowPlay. “Everyone misses poker night, and trivia night with their friends, family and neighbors. With Live Game Night, we wanted to mirror real life as much as possible to start giving people back some of their favorite social experiences. We’re confident our technology will help our players feel like they’re staring at a poker face across the table - not across a screen.”

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To host a poker game, players simply visit the Live Game Night website to sign up and download Live Poker Night to their desktop. From there, the host creates an invite link they can use to invite up to five friends to play alongside them. Live Poker Night is the first of many planned social games under the Live Game Night banner, with FlowPlay planning to launch Bunco and Mahjong over the next 90 days.

While Live Game Night offers housebound players a social outlet, FlowPlay is also working to support those on the front lines of the COVID-19 pandemic. During the month of May, players within Vegas World can donate directly to the American Cancer Society (ACS)’s Hope Lodge program, which has responded to the COVID-19 crisis by providing rooms for health care providers to rest between shifts or stay overnight. FlowPlay has raised more than $100,000 for the ACS in the past year, and encourages contributions here.

Start playing at: https://livegamenight.com/.

About FlowPlay
FlowPlay is the company behind the most powerful immersive gaming platform for casual and sports wagering games, virtual worlds and social casinos. A developer, operator and publisher, FlowPlay serves both consumers and businesses with engaging online and mobile free-to-play experiences. Partners turn to FlowPlay for its extensive multiplayer infrastructure, industry-leading monetization rates and proven track record helping businesses launch custom-branded online games. FlowPlay’s consumer products include ourWorld, one of the most popular teen virtual worlds, Vegas World, the industry’s most engaging multiplayer social casino, and Casino World, the company’s most grandiose, interactive and community-driven free-to-play social casino with tycoon gameplay. Together, they have been played by a loyal community of more than 75 million users around the world. FlowPlay dominates a cross-section of industries including online and mobile games, fantasy sports and casinos – consistently creating opportunities in high-growth and emerging markets. Based in Seattle, FlowPlay was founded in 2006 and is primarily funded by Intel Capital and the creators of Skype. For more information, visit www.FlowPlay.com or contact bizdev@flowplay.com.

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SOURCE FlowPlay