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How many wins do I need to get from Diamond 3 to Master?

15 wins

With a 55% win rate, you would need to play 150 games, you would win 83 of them and lose 67.

With a 60% win rate, you would need to play 75 games, you would win 45 of them and lose 30.

With a 65% win rate, you would need to play 50 games, you would win 33 of them and lose 17.

With a 70% win rate, you would need to play 38 games, you would win 27 of them and lose 11.

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.