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How many wins do I need to get from Bronze 3 to Master?

100 wins

With a 55% win rate, you would need to play 1000 games, you would win 550 of them and lose 450.

With a 60% win rate, you would need to play 500 games, you would win 300 of them and lose 200.

With a 65% win rate, you would need to play 333 games, you would win 216 of them and lose 117.

With a 70% win rate, you would need to play 250 games, you would win 175 of them and lose 75.

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.