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How many wins do I need to get from Silver 2 to Gold?

8 wins

With a 55% win rate, you would need to play 80 games, you would win 44 of them and lose 36.

With a 60% win rate, you would need to play 40 games, you would win 24 of them and lose 16.

With a 65% win rate, you would need to play 27 games, you would win 18 of them and lose 9.

With a 70% win rate, you would need to play 20 games, you would win 14 of them and lose 6.

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.