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Intresting combination & probability problem

I am a DOTA 2 player and I came across this intriguing problem which I do not know how to solve. In this game 1 match is played between two teams, each consisting of 5 players. Each player has to pick a hero/character from a pool of total 107 heroes, i.e 10 heroes picked. Moreover both team's captain bans 5 heroes for other team to pick. i.e 10 heroes banned.

Now, in a tournament of 16 teams, played on a knockout format with each contest between two teams consisting of 3 matches (best of 3 winner) and grand final of 5 matches. What is the probability that atleast X (lets say 5) heroes will not be picked or banned in whole tournament.

Note by Musabbir Hussain
2 years, 8 months ago

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I don't know of any simple formula for this one, but here's what I came up with:

In the whole tournament, there will be \( (8+4+2) \cdot 3 + 1 \cdot 5 = 47 \) matches. That translates to \( 47 \cdot 20 = 940 \) hero-selections (either for picking or banning, that isn't our concern).

If you don't want \( y \) players to be selected in the whole tournament, then, effectively in every match you have \( \displaystyle N_y = 107 - y \) heroes. So, in each match, you can have \(\displaystyle {N_y \choose 20} \) selections. And since there are \( 940 \) matches, totally there are \( \displaystyle {N_y \choose 20}^{940} \) possible selections.

Thus, the probability of \( y \) heroes being completely neglected in the whole tournament is \(\displaystyle P(y) = \dfrac{ {N_y \choose 20}^{940} } { {107 \choose 20}^{940} } \).

So, the probability that at least \( X \) are totally neglected in the whole tournament is:

\( \displaystyle 1 - \sum_{0 \le y \le {X-1} } P(y) \) Parth Thakkar · 2 years, 8 months ago

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@Parth Thakkar Thank you Parth, But i think ( 8 + 4 + 2 ) x 3 + 1 x 5 = 47...So, there will be 47 matches in total 15 contests..that would mean 47 x 20 = 940 hero-selections. Right?

A good solution though. Musabbir Hussain · 2 years, 8 months ago

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@Musabbir Hussain Aah that was so dumb! :D Thanks for pointing it out. Dumb Dumb Dumb! Sometimes its fun to do dumb things though ;) :D Edited. Parth Thakkar · 2 years, 8 months ago

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@Parth Thakkar No Problem..I also do such dumb things alot :D..! another thing I would like to understand is that why did u put ( 940 20 ) in denominator..Can you please explain the equation in words. I am sorry if that is something too obvious. Sadly, I am much interested but not so good in Probability, Combination & Permutation :-S Musabbir Hussain · 2 years, 8 months ago

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@Musabbir Hussain That's another mistake I made! Well, it should be \( {107 \choose 20} \) and not \( {940 \choose 20} \). That's the total number of possible selections, without any player being neglected. Parth Thakkar · 2 years, 8 months ago

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@Parth Thakkar

And since there are \(1140\) matches,

Isn't there only 14 matches. Other than that, good solution Siddhartha Srivastava · 2 years, 8 months ago

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One question. Can both teams ban the same hero? Siddhartha Srivastava · 2 years, 8 months ago

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@Siddhartha Srivastava No. A hero can be picked/banned only once. Musabbir Hussain · 2 years, 8 months ago

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