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

5 wins

With a 55% win rate, you would need to play 50 games, you would win 28 of them and lose 22.

With a 60% win rate, you would need to play 25 games, you would win 15 of them and lose 10.

With a 65% win rate, you would need to play 17 games, you would win 11 of them and lose 6.

With a 70% win rate, you would need to play 13 games, you would win 9 of them and lose 4.

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.