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How many wins do I need to get from Bronze 2 to Master?

96 wins

With a 55% win rate, you would need to play 960 games, you would win 528 of them and lose 432.

With a 60% win rate, you would need to play 480 games, you would win 288 of them and lose 192.

With a 65% win rate, you would need to play 320 games, you would win 208 of them and lose 112.

With a 70% win rate, you would need to play 240 games, you would win 168 of them and lose 72.

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.