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How many wins do I need to get from Emerald 2 to Diamond?

10 wins

With a 55% win rate, you would need to play 100 games, you would win 55 of them and lose 45.

With a 60% win rate, you would need to play 50 games, you would win 30 of them and lose 20.

With a 65% win rate, you would need to play 33 games, you would win 21 of them and lose 12.

With a 70% win rate, you would need to play 25 games, you would win 18 of them and lose 7.

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.