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.

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TopNewestI 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 · 3 years ago

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A good solution though. – Musabbir Hussain · 3 years ago

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– Parth Thakkar · 3 years ago

Aah that was so dumb! :D Thanks for pointing it out. Dumb Dumb Dumb! Sometimes its fun to do dumb things though ;) :D Edited.Log in to reply

– Musabbir Hussain · 3 years ago

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 :-SLog in to reply

– Parth Thakkar · 3 years ago

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.Log in to reply

Isn't there only 14 matches. Other than that, good solution – Siddhartha Srivastava · 3 years ago

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One question. Can both teams ban the same hero? – Siddhartha Srivastava · 3 years ago

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– Musabbir Hussain · 3 years ago

No. A hero can be picked/banned only once.Log in to reply