Quote:pwcrabbSteen suggests that "The house edge doesn't change depending on how many rolls you let bets ride!" However, Steen himself offers a calculation of per-roll loss for a Place bet on 6. Exposure for multiple rolls would seem to have consequences for the house edge, and of course it does so. His suggestion is patently silly. The theoretical probabilities of course never change, but duration of exposure can change and does so at player discretion, with attendant implications for house edge.

Expected Return for a $6 wager on Place 6 after one projected roll:

( 5 / 36 ) ( $7 ) + ( 30 / 36 ) ( $6 ) = ( $5.9722 )

Expected Loss = ( $0.0278 )

House Advantage = ( 0.4630 % )

Expect Return for a $6 wager on Place 6 after two projected rolls:

( 5 / 36 ) ( $7 ) + ( 30 / 36 ) [ ( 5 / 36 ) ( $7 ) + ( 30 / 36 ) ( $6 ) ] = ( $5.9491 )

Expected Loss = ( $0.0509 )

House Advantage = ( 0.8488 % )

The number of projected rolls is chosen at leisure. For most players of Place bets, that number is infinity

Just for grins, extending the equation above, after how many projected rolls does the H.A. closely approximate the standard presentation, which implicitly assumes exposure until resolution followed by wager removal?

The Standard Presentation is ( 5 / 11 ) ( $7 + 6 ) = ( $5.9091 )

Expected Loss = ( $0.0909 )

House Advantage = ( 1.5152 % )

I don’t know anyone that makes a place bet to infinity rolls. That would take a long time. Most players place bet until a seven out, which is very different than infinity.

Hear! Hear!Quote:unJonMost players place bet until a seven out, which is very different than infinity.

Sometimes math types consider such things as those history making rolls but who here has ever been present for one or expects to be.

Many tables are Point then SevenOut, sometimes it goes far longer. I think 8 rolls is the average but variance is high. No one needs to calculate for an infinitely long roll.

Quote:pwcrabbSteen suggests that "The house edge doesn't change depending on how many rolls you let bets ride!" However, Steen himself offers a calculation of per-roll loss for a Place bet on 6. Exposure for multiple rolls would seem to have consequences for the house edge, and of course it does so. His suggestion is patently silly. The theoretical probabilities of course never change, but duration of exposure can change and does so at player discretion, with attendant implications for house edge.

Expected Return for a $6 wager on Place 6 after one projected roll:

( 5 / 36 ) ( $7 ) + ( 30 / 36 ) ( $6 ) = ( $5.9722 )

Expected Loss = ( $0.0278 )

House Advantage = ( 0.4630 % )

Expect Return for a $6 wager on Place 6 after two projected rolls:

( 5 / 36 ) ( $7 ) + ( 30 / 36 ) [ ( 5 / 36 ) ( $7 ) + ( 30 / 36 ) ( $6 ) ] = ( $5.9491 )

Expected Loss = ( $0.0509 )

House Advantage = ( 0.8488 % )

What's patently silly is your confusing percentage lost versus actual dollar amount lost.

Yes, your actual expected dollar loss will increase with each event (that should be obvious to anyone) but your percentage expected loss relative to your wagers remains the same! It does not increase as you seem to think.

For example, if the expected loss on a Place6 bet is given as 0.463% per roll then it remains the same for each roll. All you need do is multiply this constant by the size of your bet and the number of rolls to get the expected actual dollars lost:

--- One roll = 0.463% * $6 * 1 = $0.0278

-- Two rolls = 0.463% * $6 * 2 = $0.0556 (not $0.0509 as you showed)

---- 36 rolls = 0.463% * $6 * 36 = $1

Similarly, if the expected loss on a Place6 bet is given as 1.515% per winning or losing decision (let's just call these decisions) then it remains the same for each decision. All you need do is multiply this figure by the size of your bet and the number of decisions to get the expected actual dollars lost:

--- One decision = 1.515% * $6 * 1 = $0.0909

-- Two decisions = 1.515% * $6 * 2 = $0.1818

---- 11 decisions = 1.515% * $6 * 11 = $1

Notice that the expected loss in 36 rolls is equal to the expected loss in 11 decisions. This is because on average there are 11 decisions per 36 rolls.

Regardless of how you choose to describe your expected loss, the inherent advantage held by the house does not change (and again, I'm expressing advantage as a percentage of the player's wager which is how it's typically expressed in craps.) Run some simulations if you don't believe me. It would be folly to think that a bettor could change the inherent house advantage through a scheme of betting patterns.

"Mr Boxman, how much does a $6 Place 6 pay?"

"It pays $7 every time a 6 rolls sonny." (very old boxman)

"But I'm only playing for 2 rolls now, then 3 rolls later, and maybe one roll after dinner."

"Thank you for telling me that son. Our payoffs are dependent on the number of rolls that you stay up. We have different payoffs for each bettor depending on his/her betting pattern so we appreciate finding out in advance. Hey boss, can we get a fill over here? This guy is knocking our advantage all to hell!"

Quote:pwcrabbThe number of projected rolls is chosen at leisure. For most players of Place bets, that number is infinity

They play for infinity? Wow, I guess I better play more often then. It'll take me at least a few years to hit infinity. :-)

Quote:pwcrabbJust for grins, extending the equation above, after how many projected rolls does the H.A. closely approximate the standard presentation, which implicitly assumes exposure until resolution followed by wager removal?

The Standard Presentation is ( 5 / 11 ) ( $7 + 6 ) = ( $5.9091 )

Expected Loss = ( $0.0909 )

House Advantage = ( 1.5152 % )

You asked about rolls but then gave your answer in dollars and percentage. Do you have a dollars-to-rolls conversion factor that helps answer the question? :-)

Steen

If it survives for eight at-risk rolls then Hallelujah, because the probability of that occurrence is ( 30 / 36 ) raised to the exponent ( 8 ). You may use your own calculator.

In theory that Place bet could survive without limit. The probability is always greater than zero that Big Red will not occur at that table until some far distant roll. You may choose your own large exponent and see for yourself. The probability diminishes to zero only when the exponent is infinity.

Craps bets that are not contract bets may be removed at any time. Rather than in years, their projected investment lifetimes are measured in rolls. The applicable discount factor for any particular projected future roll will be based upon the probability of not having experienced Big Red on projected prior rolls.

For an imminent next-roll future, the per-roll percentage cost of a Place bet is stated above by several forum participants. Evaluated anew after each survived roll, the next imminent roll presents an identical prospect. If the future is always only one time period away then there is no need to compound the discount factor. If the future is always only one time period away then costs and revenues remain constant and may be simply summed.

For a future period that is multiple time periods away, however, the discount factor must be compounded. Both the costs and the revenues of that future period must be discounted back to present value by multiplying them by the discount factor raised to the applicable exponent. Imminent, near future, and far future costs and revenues may then be appropriately compared.

Quote:pwcrabbThe analysis technique known as Net Present Value (NPV) is used by money managers to evaluate investment projects.

Seriously?

I state that the house edge doesn't change depending on how many rolls you let bets ride!

You reply that my suggestion is patently silly and now offer up "Net Present Value" as your proof? Nonsense.

House Advantage has been well defined and used for hundreds of years. It's not invalidated nor supplanted by Net Present Value! HA is a distillate of probabilities and outcomes which provides a fundamental metric for evaluating bets. Does it tell you everything? Of course not. There are many other metrics you can use to analyze bets but none invalidate HA.

HA represents a probable future value of a present wager, not the probable present value of a future wager.

Knock yourself out if you feel NPV adds something meaningful to your craps analysis but I've not seen anyone else use it so I can't imagine you'll have much company. I for one would be interested to know what nuggets of wisdom you've found with it.

Steen

Quote:DeMangoSteen, do you know what you can’t fix?

I'm all ears.

LOL. not even close on the evQuote:pwcrabbExpect Return for a $6 wager on Place 6 after two projected rolls:

( 5 / 36 ) ( $7 ) + ( 30 / 36 ) [ ( 5 / 36 ) ( $7 ) + ( 30 / 36 ) ( $6 ) ] = ( $5.9491 )

Expected Loss = ( $0.0509 )

House Advantage = ( 0.8488 % )

not even close on the HE

Place bet 6 per roll ev = 5/36*7 + 6/36*-6 + 25/36*0 = -1/36

HE = ev/bet resolved(action or handle) = -1/36 / $6 = -1/216

1 roll is fineQuote:pwcrabbJust for grins, extending the equation above, after how many projected rolls does the H.A. closely approximate the standard presentation, which implicitly assumes exposure until resolution followed by wager removal?

ev 1 roll = -1/36

HE = -1/36 /$6 = -1/216

ev 2 rolls = -2/36

HE = -2/36 /$12 = -1/216

ev 3 rolls = -3/36

HE = -3/36 /$18 = -1/216

all 9 possible outcomes for 2 rolls

event | prob | handle | edge | ev |
---|---|---|---|---|

w,w | 0.019290123 | 12 | - 1/216 | -0.001071674 |

w,l | 0.023148148 | 12 | - 1/216 | -0.001286008 |

w,t | 0.096450617 | 12 | - 1/216 | -0.005358368 |

l,w | 0.023148148 | 12 | - 1/216 | -0.001286008 |

l,l | 0.027777778 | 12 | - 1/216 | -0.00154321 |

l,t | 0.115740741 | 12 | - 1/216 | -0.006430041 |

t,w | 0.096450617 | 12 | - 1/216 | -0.005358368 |

t,l | 0.115740741 | 12 | - 1/216 | -0.006430041 |

t,t | 0.482253086 | 12 | - 1/216 | -0.026791838 |

total >> | 1 | . | total >> | -0.055555556 |

. | . | . | edge >> | -0.00462963 |

all 27 possible outcomes for 3 rolls

event | prob | handle | edge | ev |
---|---|---|---|---|

w,w,w | 0.002679184 | 18 | - 1/216 | -0.000223265 |

w,w,l | 0.003215021 | 18 | - 1/216 | -0.000267918 |

w,w,t | 0.013395919 | 18 | - 1/216 | -0.001116327 |

w,l,w | 0.003215021 | 18 | - 1/216 | -0.000267918 |

w,l,l | 0.003858025 | 18 | - 1/216 | -0.000321502 |

w,l,t | 0.016075103 | 18 | - 1/216 | -0.001339592 |

w,t,w | 0.013395919 | 18 | - 1/216 | -0.001116327 |

w,t,l | 0.016075103 | 18 | - 1/216 | -0.001339592 |

w,t,t | 0.066979595 | 18 | - 1/216 | -0.005581633 |

l,w,w | 0.003215021 | 18 | - 1/216 | -0.000267918 |

l,w,l | 0.003858025 | 18 | - 1/216 | -0.000321502 |

l,w,t | 0.016075103 | 18 | - 1/216 | -0.001339592 |

l,l,w | 0.003858025 | 18 | - 1/216 | -0.000321502 |

l,l,l | 0.00462963 | 18 | - 1/216 | -0.000385802 |

l,l,t | 0.019290123 | 18 | - 1/216 | -0.00160751 |

l,t,w | 0.016075103 | 18 | - 1/216 | -0.001339592 |

l,t,l | 0.019290123 | 18 | - 1/216 | -0.00160751 |

l,t,t | 0.080375514 | 18 | - 1/216 | -0.00669796 |

t,w,w | 0.013395919 | 18 | - 1/216 | -0.001116327 |

t,w,l | 0.016075103 | 18 | - 1/216 | -0.001339592 |

t,w,t | 0.066979595 | 18 | - 1/216 | -0.005581633 |

t,l,w | 0.016075103 | 18 | - 1/216 | -0.001339592 |

t,l,l | 0.019290123 | 18 | - 1/216 | -0.00160751 |

t,l,t | 0.080375514 | 18 | - 1/216 | -0.00669796 |

t,t,w | 0.066979595 | 18 | - 1/216 | -0.005581633 |

t,t,l | 0.080375514 | 18 | - 1/216 | -0.00669796 |

t,t,t | 0.334897977 | 18 | - 1/216 | -0.027908165 |

total >> | 1 | . | total >> | -0.083333333 |

. | . | . | edge >> | -0.00462963 |

I see same HE at each roll because the probabilities never change and the payoff never changes (for a unit wager). Others that use their math method may show something way different.