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How many wins do I need to get from Emerald 4 to Master?

40 wins

With a 55% win rate, you would need to play 400 games, you would win 220 of them and lose 180.

With a 60% win rate, you would need to play 200 games, you would win 120 of them and lose 80.

With a 65% win rate, you would need to play 133 games, you would win 86 of them and lose 47.

With a 70% win rate, you would need to play 100 games, you would win 70 of them and lose 30.

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.