1.414 passline

What does the 'pr' stand for in this formula?

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%.

Quote: sanstar

What does the 'pr' stand for in this formula?

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

pr = Probability

I.E.: pr(7) = probability that a roll will be a 7 = 6/36

oops
double post

error

p() is sort of generic, and can mean anything.

Here is a time saver if you haven't figure it out. Write out those long expressions in Excel
=(3/36)*(3/9) + (4/36)*(4/10) + (5/36)*(5/11) + (5/36)*(5/11) + (4/36)*(4/10) + (3/36)*(3/9) +8/36

Then format the output to be a fraction of up to three digits. You will get the correct answer of 244/495 displayed in the spreadsheet. It saves you a lot of busy work.

AFAIC, the most useful aid for the pass/come bets is the "perfect 1980", below:

result	ways	comment
comeout win	440
comeout loss	220	660 comeout decisions
win on 6	125
loss on 6	150
win on 8	125
loss on 8	150
win on 5	88
loss on 5	132
win on 9	88
loss on 9	132
win on 4	55
loss on 4	110	1320 point decisions
win on 10	55	536 point wins
loss on 10	110	784 seven-outs
	----
	1980

You can learn a lot about craps by studying that table, which has 976 winning results, 1004 losing ones. 1980 is the smallest number allowing each distinct outcome to be expressed as an integer. Of course, 976/1980 gives .42929..., the probability of winning a pass/come bet, and if you add up all the wins and losses, you get -28, the expected loss from 1980 bets, and 28 / 1980 = .01414.... The probability of a seven-out on any bet is 784 / 1980 = .395959..., so you can figure that the average length of a hand is 3.375 / .395959 = 8.52. (The average number of rolls in a decision is 3.375, which is a bit more complicated to figure.)
Cheers,
Alan Shank