I'm currently at work but when I get off I will happily work on writing up a formula for this but it is far more complex than what you have here.

I lied I read your initial conditions wrong this is a simple problem.
Each match is an independent even thus let X =win and we get the following

P(X)= (1/2)^12= .000244 w 50/50 odds
Thus
X^12=1/2 if we want 50% chance of X winning 12 indep events
Thus X is approx .9438

Thus you must win at least 94.39% of matches to guarantee an arena win % of greater than or equal to 50%.


I think you made this too hard when you thought about it, since they are all independent evens this solves out easily algebraically.

This post was edited by Xx Shin3d0wn xX on Oct 15 2015 07:26am

That's not quite right. x^12 = 1/2 is the probability of an event required to have a 50% chance of getting that event consecutively 12 times with no breaks. He has, if I understand correctly, 2 losses permitted. So what we really need is a negative binomial distribution. You'd need to write out the probabilities of winning in 12 games (which you already found 0.000244), 13 games and 14 games and get them to add up to 50%.
The probability of nth trial being kth success is [(n-1) choose (k -1)] * p^k * (1−p)^(n−k) and we are trying to figure out p. So you end up with

[p^12] + [(12 choose 11) * p^12*(1-p)] + [(13 choose 11) * p^12 * (1-p)^2] = 0.5

Solving that for p would give the true win rate required to win half of all best-12-out-of-14 matches

p^12 + 12p^12 - 12p^13 + 78p^12 * (1 - 2p + p^2) = 0.5
13p^12 - 12p^13 + 78p^12 - 156p^13 + 78p^14 = 0.5
78p^14 - 168p^13 + 91p^12 = 0.5

which is what he actually started with. Going back to one of his posts, he did the right thing by looking at the 3 possibilities and summing them, fortunately he had fairly simple cases so he didn't need to know a general distribution formula to calculate them.
I didn't understand his question initially regarding the 12/14 wins to secure an arena win.

Oh yeah, and the way I found the answer for you earlier is just graphing the equation. Since you don't need an exact answer anyway, it's much easier to essentially interpolate the approximate solution. It would be fairly trivial to write a script that would very quickly approximate a solution and would get it more and more accurate over time. If you are happy with like 2-3 digit accuracy, just toss the equation into something like http://www.mathsisfun.com/data/grapher-equation.html and zoom in on the intersection with the x-axis.

edit: here's a link to that same place with your specific formula and zoomed in. That's neat that they allow you to link to it with URL params.
http://www.mathsisfun.com/data/grapher-equation.html?func1=156x%5E14%20-%20336x%5E13%20+%20182x%5E12%20-1%20=%20y&xmin=0.8135&xmax=0.8136&ymin=-0.00004008&ymax=0.00003449
here's a place that will automatically do it for you and just display an approximate result:
www.mathportal.org/calculators/polynomials-solvers/polynomial-roots-calculator.php?val1=156x^14+-+336x^13+plus+182x^12+-1&val2=0&val3=1
You put in a formula and give xmin as 0 and xmax as 1 and you get your result

This post was edited by russian on Oct 15 2015 12:41pm

Agree with 78X^14 - 168X^13 + 91X^12 = 0.5, if I understand the problem correctly.
This equation can not be solved algebraically, as any general polynomial equation with degree 5 or more.

Looking for an approximate solution, with a graph or any other method, is the way to go.

@russian
@feanur

Thanks for the help guys, especially russian for linking what you did.

I love trying to solve (for me) challenging mathematical problems, without the knowledge of how to do so.
Discovering the process, getting closer to the answer (and sometimes failing), suddenly getting a "major" breakthrough (like aha, so that's how it works!) is part of what makes it fun for me.

But I guess it makes sense that I couldn't get any closer to a way of solving that equation, since it is impossible then!
I just figured there probably was a way, and wanted to try.