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

108 wins

With a 55% win rate, you would need to play 1080 games, you would win 594 of them and lose 486.

With a 60% win rate, you would need to play 540 games, you would win 324 of them and lose 216.

With a 65% win rate, you would need to play 360 games, you would win 234 of them and lose 126.

With a 70% win rate, you would need to play 270 games, you would win 189 of them and lose 81.

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.