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Win rate and luck
Ohoy people!
I don’t know if this has been discussed earlier on the forum since you can’t search for words shorter than four letters, hence “win rate” is out of the question. Anyways, I was thinking about the connection between luck and win rate and the fact that people often are discussing if it is based on luck or skill. Just for my own amusement I decided to calculate approximately how much luck affects your win rate and I thought I might share the results here.
First, a few points, if you plan to just trust me, don’t click the spoiler tag:
- I assume that you keep a constant expected win rate, which is kinda unrealistic since you probably get better at the game while playing it, higher tiers are more difficult etc. but that would be a pain to model.
- The more games you play the better your win rate will reflect your actual win rate, this is know as the law of large numbers, http://en.wikipedia.org/wiki/Law_of_large_numbers.
- I use a 95% conficende level in my calculations.
- I will add all the calculations i did in a spoiler tag at the end, you can run them yourself at WolframAlpha.com
- Edit: For some of you, especially if you have studied statistics, all this might seem trivial, but please remember that WoWs have players ranging from people with university level statistics education to grade school pupils that only stopped believing in Santa a few years ago.
- Edit 2: There are many ways of increasing your win rate, like playing strong ships or playing in divisions. This is however not based on luck. This post discusses the improbability of lucking your way to a high win rate, and that no-one is going to get a significantly higher win rate than somebody else based on luck. This post does not explain why a player get a specific win rate, only that it is impossible to get a high WR by pure luck.
Assuming that you have a true win rate of 50% this table tells you within what area most players (95% percent of the to be exact, 2,5 will have higher and 2,5 will have lower) win rates will end up based on luck alone.
Remember that this does not mean that players win rates will be evenly distributed within the given interval, most will cluster around the actual value since this is a normal distribution:
Just for even more fun, I decided to assume that win rate actually is based on pure luck, that is that we all actually should have a win rate of 50%, some of us just get good MM and plenty of enemy detonations, and then calculate how lucky you would need to be to achieve some of the win rates we see in game, starting out with my own. I have a win rate of 58,88% in 2218 games, which is on the high side, but in no way extraordinary.
The chance of someone winning 1306 games (or more) out of 2218 is approximately 2*(10^-17), or 0,000000000000002%. This might seem improbable compared to winning the jackpot in a large lottery which is 3*(10^-9) or 0,0000003% but it still isn’t outside the realm of possibilities, so lets go have a look at someone reaaally good: _FTD_.
According to Warships today _FTD_ have played 8010 matches and won 6339 of them. Since we all know that win rate is mere luck and _FTD_ is not better at this game, just luckier, let’s calculate exactly how lucky he/she is:
The chance for reaching _FTD_’s score just by chance i 7*(10^-623). I won’t write that out since its basically a 7 preceded by 623 zeros. To put that in perspective let’s say you have a sand box, or better yet, take the Sahara desert. You randomly pick on grain of sand, mark it, throw it back in the desert and let a friend pick a random one, and trust luck that he will get the same one as you. This might seem improbable, but it is child’s play compared to _FTD_’s luck. To approximate that, we need to go bigger, MUCH bigger. So assume that you pick an atom somewhere in the observable universe and get _FTD_ (Edit: make sure that his/her luck bringing friends are also there) to guess which one you picked, when he/she has guessed correctly 7 times in a row, that would still be more than a hundred billion times as likely as getting his/her win rate based on pure luck. So _FTD_, if you are reading this, stop playing WoWs and go buy some lottery tickets, you are vasting your time!
I now realize that what started out as a fairly serious post kinda went of the rails, but my main point is that after a good number of games, you can expect luck to be responsible for about +-1 percentage point of your win rate. The table can also be used for ballparking the values for other win rates as well even if it wont be exact. For instance, after 3000 games with a 60% WR the probable diff would be 1,75% whilst it is 1,79% for a true WR of 50%. Generally the further away from 50%, the smaller the deviation will be, but as I said, ballpark.
And if someone was planning on answering this post with “but there are a lot of players and someone has to be really lucky”, then pose yourself the question: There are plenty of people in the world, how many of them do you think would be able to guess the correct grain of sand in the Sahara desert, or a particle in the universe? There are not more players than there are people in the world and if no one in the world can do something that is much much much more likely than lucking their way to _FTD_’s win rate, no player is going to be that lucky.
Have a smiley for getting to the end of the post 
WolframAlpha code:
For calculating the interval of win rate where n is the number of games, 0.95 confidence
binomial confidence interval n=50, p-hat=0.50
Probability of getting my win rate, or above, by pure luck
P(X>1306) for X~binomial n=2218 p=0.50
Probability of getting _FTD_’s win rate, or above
P[X>6339] X~binomial n=8010 p=0.50
