as gamblers we want the variance to be as high as possible?
Depends on what you are after. a bet on a group of 18 gives you the best chance of winning, but has the smallest payout.
Betting on every number, including 0, has the best chance of winning - you 'win' every spin. It's just that you only win 36 units but pay 37 units.
Doing that gets you the expected result of losing one 37th of your stake, every spin.
As a gambler, you know that in the long run you're going to lose 1/37th of everything you bet. Your hope is to win big early on, and quit. That's your only chance of coming out ahead.
Betting on every number makes "the long run" very short. It has zero variance, and so you have no chance of "beating the odds".
I calculated the variance for each of the roulette bets: covering a single number, a pair of numbers, 3 numbers, etc.
It seems to me that the less numbers you cover, the higher the variance, and so the more chance you have of being ahead at some point.
cover payout chance expect variance
----- --------- -------- -------- --------
1 36.00000x 2.703% -0.02703 34.08035
2 18.00000x 5.405% -0.02703 16.56684
3 12.00000x 8.108% -0.02703 10.72900
4 9.00000x 10.811% -0.02703 7.81008
6 6.00000x 16.216% -0.02703 4.89116
12 3.00000x 32.432% -0.02703 1.97224
18 2.00000x 48.649% -0.02703 0.99927
36 1.00000x 97.297% -0.02703 0.02630
37 0.97297x 100.000% -0.02703 0.00000
Here's how I came up with that table:
#!/usr/bin/env python
targets = [1, 2, 3, 4, 6, 12, 18, 36, 37];
print '%5s %9s %8s %8s %8s' % ('cover', 'payout', 'chance', 'expect', 'variance')
print '%5s %9s %8s %8s %8s' % ('-----', '---------', '--------', '--------', '--------')
for target in targets:
payout = 36.0 / target
chance = target / 37.0
win_profit = payout - 1
lose_profit = -1
average_profit = chance * win_profit + (1-chance) * lose_profit
variance = ( chance * ( win_profit - average_profit)**2 +
(1-chance) * (lose_profit - average_profit)**2)
print '%5d %8.5fx %7.3f%% %8.5f %8.5f' % (target, payout, chance*100, average_profit, variance)
but still doesn't work in dice 
