Winner-Take-All Sit 'N Go Tournament Model
The Mathematica code below is designed to model the results of playing many sit 'n go poker tournaments.

Assumptions
1. Hero plays only winner-take-all tournaments with a prize pool of $1 from each player.
2. Hero has sufficient bankroll to lose every tournament she enters.
3. Each villain has an identical chance of winning each tournament.
4. In each tournament, the hero is ω times as likely to win as each villain is where ω is a constant real number.
5. Each tournament result is independent of previous results: players never tilt, nor do they learn from previous tournaments.

Variables
p = probability of hero winning any given tournament
x = probability of any one villain winning any given tournament
n = number of players per tournament
g = number of tournaments entered
c = cost of entering tournament (including rake)
w = number of wins

Helpful combinations of variables
ω = "skill factor"= (p/x)
β = "bad beat factor" = ((number of villains)/(win factor)) =(( n - 1)/ω)

Results to be derived
V[g, ω] = expectation value of g tournaments
W[w,g, ω] = probability that hero wins w games out of g

1-tournament probability of winning
Normalization of probabilities requires that exactly one person win the tournament:
ω x + (n-1) x = 1   =>   x = (1/(ω + n - 1))  =>   p = (ω/(ω + n - 1))
Therefore p depends only on the number of players n and skill factor ω .
 
  p = (1/(1 + β))   and   (1-p) = (β/(1 + β))
 
A helpful way to remember this is "Bad beats are (villains/skill) and hero will win 1 tournament for every β bad beats."   
For example, a hero who is "twice as good" as her opponents at a 10-person table has 9 villains over 2 skill.  She expects to win once for every 4.5 losses.


Expectation value of tournaments
The expectation value V[1, ω] of a single tournament depends on n, ω, and the cost c of entering a tournament:  V[1, ω] = (n-1)p - cg
According to assumption 5, the tournaments are independent.  Therefore  V[g, ω]  =  g V[1, ω] .  The expectation value for our hero after g games is

V[g, ω]  =  g (((n-1)/(1 + β))-c)  =  g (n-1)((ω  -  c)/(ω  + n - 1))

c=1.1;
V[g_,ω_, n_] := g (n-1)(( ω-c)/(ω+n-1));
Manipulate[  Plot[V[g,ω, n]  ,{ω,0.5, 3}, AxesLabel->Automatic], {g,1,100}, {n,2,100}]

Probability of winning w times after entering g tournaments
According to assumptions 1 and 5, the total number of wins after g games is a random variable determined by a sequence of independent yes/no trials.
Therefore, for any fixed ω, the distribution W[w, g, ω] must be a binomial (or Bernoulli) distribution for g trials, each with success probability p :

 W[w,g, β]  =  (({{g}, {w}}))(p^w)(((1-p))^(g-w))  =  ((g!)/(w!(g-w)!))((((1/(1+β))))^w)((((β/(1+β))))^(g-w))  =  ((g!)/(w! (g-w)!))((((β/(1+β))))^g)(β^(-w))

W[w_, g_, n_, ω_] := Binomial[g,w] * ((((((((n-1)/ω)))/(1+(((n-1)/ω))))))^g)(((((n-1)/ω)))^(-w))

Manipulate[   ListPlot[ Table[{w,W[w,g,n, ω]},{w,0,g}] , PlotRange->All, AxesOrigin->{0,0},PlotMarkers->{Automatic, Medium} ] ,
{g,1,500}, {n,2,100}, {ω, 0.1, 5}   ]

Probability of winning m games or *fewer* is given by L-function; probability of winning m games or *more* is given by M-function:

L[m_, g_, n_, ω_] := (∑_(w=0)^m)Binomial[g,w] * ((((((((n-1)/ω)))/(1+(((n-1)/ω))))))^g)(((((n-1)/ω)))^(-w)) ;
M[m_, g_, n_, ω_] := (∑_(w=m)^g)Binomial[g,w] * ((((((((n-1)/ω)))/(1+(((n-1)/ω))))))^g)(((((n-1)/ω)))^(-w));
Manipulate[   ListPlot[ Table[{m,L[m,g,n, ω]},{m,0,g/2}] , PlotRange->All, AxesOrigin->{0,0},PlotMarkers->{Automatic, Medium} ] ,
{g,1,1000}, {n,2,100}, {ω, 0.1, 10}   ]
Manipulate[   ListPlot[ Table[{m,M[m,g,n, ω]},{m,0,g/2}] , PlotRange->All, AxesOrigin->{0,0},PlotMarkers->{Automatic, Medium} ] ,
{g,1,1000}, {n,2,100}, {ω, 0.1, 10}   ]