> If the sample is x standard deviations from expected, what
> chance of a gaffed machine does that translate to?
>
This will depend on
(1) before any data, how likely do you believe the machine is biased
e.g. Before the test, you thought there was a 1% change of a gaffed
machine
The test tells you afterwards that there is a 30-40% chance of a
gaffed machine.
Your question is very difficult because it relies on your "a priori"
estimate of how likely a bad machine is...
(a) For Las Vegas, I would set this very low.
(b) For a machine built by a non-standard company in a foreign
country or cruise ship, maybe higher.
Especially if other people also report low numbers of quads.
(2) how badly-biased you believe the machine to be
e.g. a 99% machine (Fair = "Null Hypothesis") vs. 96% machine (96%
machine = Hypothesis A)
A machine that has too many non-quads, might be modeled by a "Zero-
Inflated Poisson" distribution;
e.g. 75% of no quads, 25% of a legitimate VP machine.
http://data.princeton.edu/wws509/stata/overdispersion.html
(3) Use t-tests or Bayesian reasoning or etc... to test hypothesis
and evaluate beliefs.
http://en.wikipedia.org/wiki/Statistical_hypothesis_testing
http://www.stats.gla.ac.uk/steps/glossary/hypothesis_testing.html
In reality, since I don't know if a gaffed machine pays 85%, 90%, 95%,
etc...
(1) I would be really lazy and see if I was +/- 2-3 SDs out.
In Vegas I would not worry unless I was >3 SDs below normal.
In an unreliable area, I might even start worrying if I was >1 SD
below normal (e.g. maybe play a cheaper denomination as a test?)
In probability langauge,
(A) Test just the "Null Hypothesis" (Machine is Fair) with a 5% or 1%
significance.
(B) If your result is outside the 5%/1% significance area (e.g. 2-3
SDs), then maybe the machine is not fair?
+/- 3 SDs for a normal distribution is 0.3% likely.
...however (my statistics is rusty, apologies if I'm messing this
up), for getting Quads, a Bernoulli or Poisson approximation is more
relevant
http://en.wikipedia.org/wiki/Poisson_distribution
I forget what 3 SDs means for a Poisson distribution.
Mitchell
