I am in a contest with 200 total participants. There will be 5 total scores to determine the top 20 finishers. Each is initially ranked according to their ranking from a previous contest. That previous ranking is your first score. Mine is 193. The best starting rank is 1. The worst starting rank is 200. There will be 4 additional ranked contests that contribute to my total score. My total score will be the sum of my ranked performances from each of these 4 contests and my previous score of 193. So, if I got first place in each of the 4 additional contests, my total score would be 197. The 20 lowest scores overall move on to the next round. What is the probability that I can make the top 20 after the next 4 contests with my starting score of 193 relative to the other contestants ranging from 1-200 in their initial scores?
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The question can't be answered without knowledge of the distributions. Also, it seems unlikely that the contest results are independent. If the contests are all related somehow, say they're all math and science contests, we would expect the scores in the first contest to be good predictors of the scores in the others. – saulspatz Apr 17 '18 at 21:02
1 Answers
Let's assume the contest rankings are ransom and independent - a very bold assumption! Then again, this boldness allows us to make very crude approximations in what follows.
For the sum of five uniform random vars, it is a bit of a stretch, but let us just do as if we can approximate the sum as a normally distributed variable. We use the mean $\frac{1+200}2\approx 100$ and variance $\frac1{12}(200-1)^2\approx 3333$ for a single contest, hence mean $\approx 500$ and standard deviation of $\sqrt{5\cdot 3333}\approx 129$ for the sum of five contests. The top ten percent ought to exceed the mean by at least $1.28$ standard deviations, i.e., the winners will have rank sums up to $500-1.28\cdot 129\approx 335$.
To be in that range, your rank sum of four contests must be $<142$. Similar to above, the rank sum of four contests has mean $\approx 400$ and std.dev. $\approx 115$. Thus you need to be better by $\frac{400-142}{115}\approx 2.2$ sigmas, which has probability $\approx 1.4\%$.
As mentioned, this is a very rough estimate, but shows how hard it is to raise to the top be chance.
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How would the probability of being in the top 10% out 200 change if starting with a ranking of 193 as my first score, I was able to be ranked 1st for the next four contests. With my total score of 197, what would the new rough estimate be of being in the top 20, assuming the other 199 scores are random and independent? – S. Tone Apr 17 '18 at 22:52