In a typical CSGO game, 2 teams play each other for a best out of 30 (First team to 16 wins if a team reaches 16 wins). My question is what is the probability a CSGO game goes 30 games and wanted to see if my solution was correct.
I will consider $p=\text{probability team A wins}$, $q=\text{probability team B wins}$
The only way to reach 30 games is either a tie (15-15) or if the match comes to (14-15) on the 29th round and then turns to (14-16). Likewise the match could also be (15-14) on the 29th round and become (16-14)
So I was thinking for the 15-15 situation it would be 30C15$*p^{15}q^{15}$
For the 15-14 situation I figured 29C15$*p^{15}q^{14}*p$
For the 14-15 situation I figured 29C15$*p^{14}q^{15}*q$
I then took these 3 probabilities and added them for roughly a 29% chance of happening (this is if we assume $p=q=.5$)
Is this correct?
You can avoid splitting it in three cases and doing it just 2. I mean if we know that there hasn't been a winner after 29 games, the result in the last one doesn't matter too much. Hence you need to only find when the results are $15-14$ or $14-15$ after 29 rounds.