# Changing the shape of the graph of two million pass sessions

Discussion in 'Advanced Craps' started by goatcabin, Feb 17, 2010.

1. goatcabin, Feb 17, 2010

### goatcabin Member

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The symmetry of the bell curve recording the results of two million people playing the pass line for two hours each is the result of all of them playing for the full two hours.

Suppose those two million people started out with only ten times their unit bet? (There was a person posting here who plays with \$100 and could only find a \$10 table.) If that were the case, something over 400,000 of them would be expected to lose their 10 units before 200 rolls, some of them well before that. So there's going to be a big "spike" at -10 units, cutting off the left side of the graph there, while the right side, to the "good" of the expected loss, is going to look pretty much the same. The casinos are not going to win as much, because the unlucky players who lost 30 or more units in the first experiment could only lose 10 in this one. This means less total money was bet, and the less money bet the less the expected loss. We say this graph has "positive skew", because the right tail is longer than the left one. The other interesting thing is that the players who last the two hours are actually going to have an average win, because none of them lost as many as 10 units. It's sort of like life expectancy: as you grow older, your life expectancy increases, because everyone who has died by that age is no longer part of the "universe".

Now suppose we told those two million people with their 10-unit bankrolls to establish a 10-unit win goal. If they get 10 units ahead, they put 8 of them aside and play with 2, trying to win 5 more, but stopping if they lose the 2. They can keep going as long as they don't lose 20% of what they've won. Now we are going to "bump up against" peaks on both sides of the bell. We're still going to have over 20% who bust, but now we'll have some 12% who end up 8 units ahead, having gotten up 10 and then lost the 2. Others will not lose back the 2 and go on to win more. If you reach a win goal and decide to risk 20% of it, it's like starting play with a very short bankroll. If you lose it, you stop, but, if not, you keep playing until you reach the next win goal, again setting aside most of it and continuing to play with a couple of units. In rare cases, this can go on for several goals.

I ran a simulation of this, using a program I wrote. The first win goal is 10 units, so the bankroll would be 20. Upon reaching 20, set 18 aside, play with 2 and set the next goal at 25 (+15). Upon reaching 25, set 23 aside, play with 2 and set the next goal at 30, i.e. each time try for another half of the original win goal. Each time, it's like starting a new session with just two units and trying to win five units. It's not very likely, but the idea is that you don't want to lose back much of what you've won. If your luck continues, you keep going; else you quit.

In order to reach a win goal starting with a short bankroll, it helps to have a lot of variance. Of course, more variance means you're more likely to lose the short bankroll. I ran a simulation of 20,000 sessions like this:

bet \$5 pass, taking 3, 4, 5X odds (avg. bet will be about \$19)
establish win goal of \$100 (bankroll \$300)
save \$70 and play with \$30, trying to win \$50 more; repeat for each win goal reached

stop when the FIRST of these occurs:
bankroll is gone
OR
you've lost back the \$30 you played with after reaching one or more win goals
OR
you've reached 200 rolls AND no bet is outstanding AND you've just lost a bet

The program keeps track of just about everything.

Here's how it came out:

bust: 25.4%
reached a win goal: 55.4%
time limit: 19.2%

Over 5000 sessions busted, so there's a very large "spike" at -200 (net).
The are smaller spikes at +70 and +75 (2208 between them), because quite a few reached the \$100 win goal (or just past it) and then lost back the \$30 and quit. Another 1900 or so sessions came out between +\$120 and +\$150.

The biggest win was \$620 in a session where nine successive win goals were reached! Six more "players" reached eight, ten reached seven and 50 reached six win goals.

Overall, there were 11,862 winning sessions, 80 broke even and 8058 lost. The mean net outcome was to lose \$2.44, and the standard deviation was \$145.

This illustrates the effect of a short bankroll (relative to the size of one's bets), a moderate win goal (half the bankroll) and pretty high volatility (from the 3, 4, 5X odds) on the shape of the graph of possible session outcomes. The method I simulated gives one a better-than-50% chance of increasing one's bankroll by 50%, or a lot more, along with about a 25% chance of losing it. "You pays your money and you takes your choice!" Whether that is desirable is up to each individual player.
Cheers,
Alan Shank

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