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

116 wins

With a 55% win rate, you would need to play 1160 games, you would win 638 of them and lose 522.

With a 60% win rate, you would need to play 580 games, you would win 348 of them and lose 232.

With a 65% win rate, you would need to play 387 games, you would win 252 of them and lose 135.

With a 70% win rate, you would need to play 290 games, you would win 203 of them and lose 87.

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.