**Ask a monkey a physics question thread**

[Savy]

I'll probably give this a think if I have time but fancy giving me a quick answer to solve this problem. Hearthstone related obviously for those that play (I just started)

If I have a win rate of x% how do I work out the likelihood of me winning a certain number of games given the conditions.

If you lose 3 games you are finished.
If you win 12 games you are finished.

[MadMojoMonkey]

I'll probably give this a think if I have time but fancy giving me a quick answer to solve this problem. Hearthstone related obviously for those that play (I just started)

If I have a win rate of x% how do I work out the likelihood of me winning a certain number of games given the conditions.

If you lose 3 games you are finished.
If you win 12 games you are finished.

You're asking your EV for the number of wins on an arena run, given your winrate.
Let X be {winrate}.
0 <= X <= 1

Option 1: Your Arena run ends due to acquiring your 3rd loss
For 3 <= N <= 14:
(N - 1) matches produced (N - 3) wins and (2) losses, in whatever order, then the Nth match was your 3rd loss.
EV(N,X) = C(N-1,N-3)*X^(N-3)*(1-X)^2 * (1-X)
EV(N,X) = C(N-1,N-3)*X^(N-3)*(1-X)^3

Option 2: Your Arena run ends due to you defeating the 12th opponent
For 12 <= N <= 14
EV(N,X) = C(N,N-12)*X^(12)*(1-X)^(N-12)


***
I plugged this into Excel and I'm getting less than perfect results, so there's something wrong in there, but I don't see it.
To clarify, for values of 0 < X < 35%, the sum of all of those results rounds to 100.00%
For 35% < X < 64% the sum of all results is over by less than 1%
The error reaches a maximum at X ~= 88%, where the sum of all probabilities is over 106%.

Given these unexplained errors in the results, I show that you need a winrate of X > 80% to have more than 50% EV at winning 12 matches.

***
If anyone sees the error in my analysis, let me know.


EDIT: error found. I forgot to account for the fact that your final match is always a win in option 2.

Correction:
Option 2: Your Arena run ends due to you defeating the 12th opponent
For 12 <= N <= 14
EV(N,X) = C(N-1,N-12)*X^(12)*(1-X)^(N-12)

The sum over both cases for all N yields 100% for any X.