# Lost first game – can I still make my mother proud?

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?

• This appears more like a math problem to me. – Ahmed Abdelhameed Dec 19 '18 at 9:19
• Or not... since it was already answered... – jafe Dec 19 '18 at 11:11
• @AhmedAbdelhameed It is a math problem, but it's one that has a surprising paradox at its heart. I think that makes it an interesting puzzle. A puzzle that can be solved using straightforward math isn't automatically not a puzzle. – Kevin Dec 19 '18 at 14:28
• Are all players entered into this tournament? If not then the "only 0.1% of all players are better" is a useless statistic. – AndyT Dec 19 '18 at 16:12
• @jafe - I don't understand your comment. For a fuller explanation of my comment, please see my longer comment under Kruga's answer. – AndyT Dec 19 '18 at 16:41

## 1 Answer

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      100
Since 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 $$0.9801*999/1099 + 0*100/1099 = 9791199/10990000 = 89.1\%$$

• Yep, this is correct! Nice work. – jafe Dec 19 '18 at 11:19
• This is known as the false positive paradox – BlueRaja - Danny Pflughoeft Dec 19 '18 at 12:45
• But what's the likelihood that the players entered into the tournament are evenly distributed across all skill ranges? Gary Kasparov doesn't play in tournaments at my local chess club, and a beginner at my local chess club doesn't play in the Chess World Cup. I suspect that any tournament the OP enters is likely to have a higher than average skill level among the entrants, hence his chances of winning are lower than you calculate. – AndyT Dec 19 '18 at 16:11
• @AndyT - in the cutthroat world of finger wrestling, you don't choose to play. You're conscripted. :-) – Kevin Dec 19 '18 at 19:29
• @AndyT You're right that that bit is an assumption, but in the absence of the OP explicitly providing that information, assuming it's random and uniformly distributed is the only sane assumption if you're trying to produce a number. But yea, OP probably should have stated we should assume that. – Shufflepants Dec 19 '18 at 20:55