What Are Your Chances Of Winning At Craps
Posted : admin On 7/28/2022- What Are Your Chances Of Winning At Craps Games
- What Are Your Chances Of Winning At Craps Machines
- What Are Your Chances Of Winning At Craps Odds
Introduction
Players wanting to know the craps odds are likely to want to know the true odds on a wager. Using this data, a player can compare the real odds with the payouts to get a new perspective on the house edge. Craps Bet – Real Odds Pass Line Bet – 251 to 244. To increase your chances of winning, and before you put your hard earned money down on the table, you should at least have a rudimentary knowledge of the odds, payouts and house edge for each of the bets on the table. Before we get into the details, let’s get a couple of things out of the way first. The casino always wins in the long run.
Bonus Craps is a set of three side bets found in craps. There are as follows:- Small — Wins if the shooter rolls every total from 2 to 6, before rolling a total of 7. Wins usually pay 30 to 1.
- Tall — Wins if the shooter rolls every total from 8 to 12, before rolling a total of 7. Wins usually pay 30 to 1.
- All — Wins if the shooter rolls every total from 2 to 12, except a 7, before rolling a total of 7. Wins usually pay 150 to 1.
Analysis
The following table shows my analysis of the Small and Tall bets, when wins pay 30 to 1. The lower right cell shows a house edge of 18.30%.
Small and Tall
| Event | Pays | Probability | Return |
|---|---|---|---|
| Win | 30 | 0.026354 | 0.790617 |
| Loss | -1 | 0.973646 | -0.973646 |
| Total | 1.000000 | -0.183029 |
In the past, some tables were known to pay 34 to 1 on the Small and Tall. These more generous odds have a house edge of 7.76%.
The following table shows my analysis of the All bet, when wins pay 150 to 1. The lower right cell shows a house edge of 20.61%.
All
| Event | Pays | Probability | Return |
|---|---|---|---|
| Win | 150 | 0.005258 | 0.788655 |
| Loss | -1 | 0.994742 | -0.994742 |
| Total | 1.000000 | -0.206087 |
In the past, some tables were known to pay 175 to 1 on the All. These more generous odds have a house edge of 7.47%.
Methodology
I've analyzed Bonus Craps three ways, as follows:
- Simulation — This is probably the easiest way. However, for mathematical purists like me, simulations are always so intellectually unsatisfying.
- Markov Chain — This method is tedious and time consuming. For the Small and Tall it would require a 6x6 transition matrix and for the All a 12x12.
- Integral Calculus — This method is surprisginly easy with the use of an integral calculator. I explain it in more depth below.
Imagine that instead of significant events being determined by the roll of dice, one at a time, consider them as an instant in time. Assume the time between events has a memory-less property, with an average time between events of one unit of time. In other words, the time between events follows an exponetial distribution with a mean of 1. This will not matter for purposes of adjudicating the bet, because event still happen one at a time.
The following is the probability any given total has NOT been rolled at least once within x units of time:
- 2 or 12: exp(-x/36)
- 3 or 11: exp(-x/18)
- 4 or 10: exp(-x/12)
- 5 or 9: exp(-x/9)
- 6 or 8: exp(-5x/36)
- 7: exp(-x/6)
Let's look at the Small bet first. The odds are exactly the same for the Tall bet.
In x units of time, the probability that the 2, 3, 4, 5, and 6 have been rolled, and the 7 has not is: (1-exp(-x/36))*(1-exp(-x/16))*(1-exp(-x/12))*(1-exp(-x/9))*(1-exp(-5x/36))*exp(-x/6) The probability that at time x, the player rolls a 7, having previously rolled every total from 2 to 6, is:
For the integration,I used this integral calculator.
Since a winning 7 could happen at any time, the probability of winning is the integral of this probability over x from 0 to infinity. Not that we need to know, but the integral is, before putting in the limits of integration is (-6*e^(-x/6)+(36*e^(-(7*x)/36))/7+(9*e^(-(2*x)/9))/2-(36*e^(-(11*x)/36))/11-3*e^(-x/3)-(36*e^(-(13*x)/36))/13+(18*e^(-(7*x)/18))/7+(12*e^(-(5*x)/12))/5+(9*e^(-(4*x)/9))/4-(36*e^(-(19*x)/36))/19-(9*e^(-(5*x)/9))/5+(12*e^(-(7*x)/12))/7)/6.
Putting in the limits of integration, the answer is 20049 / 760760 = apx. 0.02635390924864609.
Next, let's look at the All bet.
In x units of time, the probability that the 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12 have been rolled, and the 7 has not is: (1-exp(-x/36))2*(1-exp(-x/16))2*(1-exp(-x/12))2*(1-exp(-x/9))2*(1-exp(-5x/36))2*exp(-x/6) The probability that at time x, the player rolls a 7, having previously rolled every total from 2 to 6, is:
What Are Your Chances Of Winning At Craps Games
Since a winning 7 could happen at any time, the probability of winning is the integral of this probability over x from 0 to infinity. Not that we need to know, but the integral is, before putting in the limits of integration is (-6*e^(-x/6)+(72*e^(-(7*x)/36))/7+(9*e^(-(2*x)/9))/2-8*e^(-x/4)-(18*e^(-(5*x)/18))/5-(72*e^(-(11*x)/36))/11+(72*e^(-(13*x)/36))/13+(108*e^(-(7*x)/18))/7+(24*e^(-(5*x)/12))/5-(27*e^(-(4*x)/9))/4-(216*e^(-(17*x)/36))/17-10*e^(-x/2)-(72*e^(-(19*x)/36))/19+(27*e^(-(5*x)/9))/5+(144*e^(-(7*x)/12))/7+(54*e^(-(11*x)/18))/11-(72*e^(-(23*x)/36))/23-(15*e^(-(2*x)/3))/2-(216*e^(-(25*x)/36))/25-(54*e^(-(13*x)/18))/13+(8*e^(-(3*x)/4))/3+(54*e^(-(7*x)/9))/7+(72*e^(-(29*x)/36))/29-(72*e^(-(31*x)/36))/31-(9*e^(-(8*x)/9))/8-(24*e^(-(11*x)/12))/11+(18*e^(-(17*x)/18))/17+(72*e^(-(35*x)/36))/35-e^(-x))/6.
Putting in the limits of integration, the answer is 126538525259 / 24067258815600 = 0.0052577040961964420049.