Putting in the limits of integration, the answer is 20049 / 760760 = apx. 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. 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/18))*(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:įor the integration,I used this integral calculator. The odds are exactly the same for the Tall bet.

The following is the probability any given total has NOT been rolled at least once within x units of time: This will not matter for purposes of adjudicating the bet, because events still happen one at a time. In other words, the time between events follows an exponetial distribution with a mean of 1. Assume the time between events has a memory-less property, with an average time between events of one unit of time. Imagine that instead of significant events being determined by the roll of dice, one at a time, consider them as an instant in time.

Integral Calculus - This method is surprisginly easy with the use of an integral calculator.

For the Small and Tall it would require a 6圆 transition matrix and for the All a 12x12.

Markov Chain - This method is tedious and time consuming.

However, for mathematical purists like me, simulations are always so intellectually unsatisfying.

Simulation - This is probably the easiest way.

I've analyzed Bonus Craps three ways, as follows: These more generous odds have a house edge of 7.47%. In the past, some tables were known to pay 175 to 1 on the All.

Craps repeater bet