monte carlo simulation it looks as if it would be good idea to avoid the pass line--where the expected return on investment is loss but the don' pass line looks like pretty good bet or does itlooking at the standard deviationsit seems that perhaps the don' pass line is not such good bet after all recall that under the assumption that the distribution is normalthe confidence interval is encompassed by two standard deviations on either side of the mean for the don' pass linethe confidence interval is [ * * ]--roughly [- %+ %that certainly doesn' suggest that betting the don' pass line is sure thing time to put the law of large numbers to work crapssim( passmean roi - std dev don' passmean roi - std dev we can now be pretty safe in assuming that neither of these is good bet it looks as if the don' pass line may be slightly less bad howeverbecause the confidence interval [- - for the pass line overlaps with that for the don' pass line [- - ]we cannot say with confidence that the don' pass line is better bet suppose that instead of increasing the number of hands per gamewe increased the number of gamese by making the call crapssim( as shown belowthe mean of the estimated rois are close to the actual rois howeverthe standard deviations are still be high--indicating that the outcome of single game of hands is highly uncertain crapssim( passmean roi - std dev don' passmean roi - std dev one of the nice things about simulations is that they make it easy to perform "what ifexperiments for examplewhat if player could sneak in pair of cheater' dice that favored over ( and are on the opposite sides of die)to test this outall we have to do is replace the implementation of rolldie by something like def rolldie()return random choice([ , , , , , , , , , , , ]this relatively small change in the die makes dramatic difference in the oddscrapssim( passmean roi std dev don' passmean roi - std dev no wonder casinos go to lot of trouble to make sure that players don' introduce their own dice into the game