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How many wins do I need to get from Bronze 1 to Master?

92 wins

With a 55% win rate, you would need to play 920 games, you would win 506 of them and lose 414.

With a 60% win rate, you would need to play 460 games, you would win 276 of them and lose 184.

With a 65% win rate, you would need to play 307 games, you would win 200 of them and lose 107.

With a 70% win rate, you would need to play 230 games, you would win 161 of them and lose 69.

Assuming an average LP gain of +/-25 LP per game below Emerald and +/-20 LP for Emerald+ as per Riot Phroxzon's tweets here and here.