get number of participants from given probability

Pir

New Member
#1
Hi!

I have a

pool of participants - defined
number of tournaments (competitions) - defined
participants per tournament (partPerTour)

Of the participants per tournament only 1 winner (the one with highest ranking) is put forward, everyone else is returned back to the pool for the next tournament.

I have calculated the probability of a specific participant (the one with highest ranking in the whole pool) being picked.
For this I use programming (I need to code it in the end) and just loop through the following

repeat for number of tournaments and add up
tour1:[1/(pool)]+[1/(pool-1)]...+[1/(pool-participantsPerTournament)]+
....+...+
tourx:[1/(pool-x+1)]+[1/(pool-x+1-1)]...+[1/(pool-x+1-participantsPerTournament)]

(pool is reduced by 1 participant from tournament to tournament)



But I would like to also calculate how many participants I need per tournament to have a given probability of picking a specific participant met.

I assume this is probably quite a standard probability calculation, and there might be a formula, but I can't find the right one.

Any insight are appreciated!
 
#2
But I would like to also calculate how many participants I need per tournament to have a given probability of picking a specific participant met.
This isn't completely clear to me but I read it to say, for example - 100 participants. one called X - 7 tournaments T1 to T7. Say 8 participants per tournament.
T1 pick 8 participants at random from 100, T2 pick 8 from 99, then T3 ... to T7 pick 8 from 94.
What is the probability that X has been in one of the tournaments?

Probability of X NOT being picked is 92/100x91/99x...x86/94 and the probability of being picked as 1 - that. This can all be condensed into 1-92!/85!x93!/100!
Putting this into Excel terms, probability of being picked is =1-FACT(p-n)/FACT(p-n-t)*FACT(p-t)/FACT(p) where n is the number in each tournament. There is no formula to solve this for n, but n is easily found using a spreadsheet for any given p and t.