Analysis Of Skill-Increment Required To Win A Losing Quake III Arena Match

I've recently started playing Quake III Arena, and I'm already liking it a lot. I was playing a deathmatch in which there were only 2 players - me and a bot. The bot had scored 15 frags while I had scored 10 in the same time. The limit of frags was 20.

Let's mathematically analyze the increment required in my fragging/killing/playing skill (in percentage terms), to be able to win this match, to find out if I have any chance of winning this match.

I played 66.67% as good as the bot (since I scored 10 frags while he scored 15 - in same time). If I continue to play same way, I will have got 3.33 frags in the time he gets 5 - and he shall win the match. To be able to win, I need to score 10 frags in the time he scores 5 (his frag score should tend to 5, but shouldn't actually touch 5 - else he would win). This means I need to play twice as good as he has played so far, to be able to win this match. Sad! Also, compared to my own past performance, I need to play an astounding 200% better [{(10/3.33)*100}-100] to win the match.

This explains why - having reached this point - the chances of me winning this match are narrow.

However, this analysis would not have been true had the bot scored 3 frags and I had scored 2 (although even in that case I would've played 66.67% as good as him). This is because the confidence with which it can be said that I played 66.67% as good as the bot is very low for small number of frags such as 2 or 3, whereas in the case number of frags is 10 or 15, the confidence increases.

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