Probability of rolling 6's on 3 dice - adjusted gambling odds? I'm struggling to work out odds on a game that were working on. It's probably best if I write an example as I'm really not a mathematician!
I'm working on a dice game where the player bets 1 coin and rolls 3 dice. If any of the dice are a 6 we payout based on the following. 
If 1 dice is a six we pay x
If 2 are a six we pay y
and if all of them are a 6 we pay z
This seemed simple at first as the odds of rolling 1 six are 1/2, 2 sixes would be 1/14 and 3 sixes would be 1/216 so they would be our odds. 
The issue is that in theory would have to roll the dice 216 times to get the jackpot and receive the £216 payout (at £1 per bet). However, if I did do that I would also win 108 times at 2/1, 15 times at 14.4/1 and once at 216/1 (ignoring the luck factor etc).
Therefor for my £216 stake i would win a total of £648.
So what should my payouts be for 1x6, 2x6, and 3x6 (assuming 0% house edge and all that) to be fair to the player?
Thanks for you time.
Mark

The probabilities you have are wrong. Denote by $X$ the number of sixes rolled. Then the correct probabilities are:
$P(X = 0) = \left(\frac{5}{6}\right)^3 = \frac{125}{216}$
$P(X = 1) = P$ (first dice six) $ + P$(second dice six) $ + P$(third dice six) $= \frac{1}{6} \frac{5}{6} \frac{5}{6} + \frac{5}{6} \frac{1}{6} \frac{5}{6} + \frac{5}{6} \frac{5}{6} \frac{1}{6} = 3 \cdot \frac{25}{216} = \frac{75}{216} = \frac{25}{72}$.
$P(X = 2) = P$ (first dice not six) $ + P$(second dice not six) $ + P$(third dice not six) $= \frac{5}{6} \frac{1}{6} \frac{1}{6} + \frac{1}{6} \frac{5}{6} \frac{1}{6} + \frac{1}{6} \frac{1}{6} \frac{5}{6} = 3 \cdot \frac{5}{216} = \frac{15}{216} = \frac{5}{72}$.
$P(X = 3) = \left(\frac{1}{6}\right)^3 = \frac{1}{216}$
To answer your question, if you want a fair game (expectation zero profit/loss) you can pick $x,y,z$ however you want, as long as they satisfy $75x + 15y + z = 216$. Namely, on average in 216 rolls, you will get one six $75$ times, two sixes $15$ times and three sixes once. Reasonable values could be $x = 2, y = 3.4, z = 15$.
