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

25 wins

With a 55% win rate, you would need to play 250 games, you would win 138 of them and lose 112.

With a 60% win rate, you would need to play 125 games, you would win 75 of them and lose 50.

With a 65% win rate, you would need to play 83 games, you would win 54 of them and lose 29.

With a 70% win rate, you would need to play 63 games, you would win 44 of them and lose 19.

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.