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

16 wins

With a 55% win rate, you would need to play 160 games, you would win 88 of them and lose 72.

With a 60% win rate, you would need to play 80 games, you would win 48 of them and lose 32.

With a 65% win rate, you would need to play 53 games, you would win 34 of them and lose 19.

With a 70% win rate, you would need to play 40 games, you would win 28 of them and lose 12.

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.