I am one of the greatest finger wrestlers of my generation. Only 0.1% of all players are better.
When I play against someone who's worse than me, I win 99% of the time. Against someone who's better than me, I always lose.
It's the first round of a tournament. We're playing best of three games (i.e. two wins needed). I hate to say it, but I have just lost the first game! What are my chances of still making it to the second round?
Assuming that the players who are better than me always beat the players who are worse than me, what are my chances of making it to the third round?
If you play 100.000 times versus random players, how many of them would be better/worse than you, and how many would you win/lose? We can put this in a grid.
Worse Better Win 98901 0 Lose 999 100Since you lost you first match there is a $999/1099$ chance that the your opponent is worse than you, and $100/1099$ chance that your opponent is better.
If your opponent is better, your chance of winning the series is 0. If your opponent is worse, your chance of winning is $0.99*0.99=0.9801$
You total chance of winning is then $0.9801*999/1099 + 0*100/1099 = 9791199/10990000 = 89.1\%$