Re: Heads Up Matches situation
Easy enough for 3, get's to be a pain for many.
Chance of winning a best of 3 =
Chance of winning all 3 * ways to win all 3+ chance of winning 2 * ways to win 2
= (.7)(.7)(.7)*1 + (.7)(.7)(.3)*3 = .784
There may be some nice formula that I am just not familiar with for big numbers, like best out of 25, but basically you are looking at a binomial expansion (if my memory of terms is correct) and summing over the winning ways (or losing ways and subtracting this from 1).
This is very similar to a random walk. In a random walk you start at zero and can take a step in either the positive or negative direction (based on some probability, usually 50/50 but no reason it cannot be 70/30). Where you end up after n steps will follow a normal distribution.
At any rate, the more games in the match the more likely the 70% guy will win.
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