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How many wins do I need to get from Plat 1 to Diamond?

24 wins

With a 55% win rate, you would need to play 240 games, you would win 132 of them and lose 108.

With a 60% win rate, you would need to play 120 games, you would win 72 of them and lose 48.

With a 65% win rate, you would need to play 80 games, you would win 52 of them and lose 28.

With a 70% win rate, you would need to play 60 games, you would win 42 of them and lose 18.

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.