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

76 wins

With a 55% win rate, you would need to play 760 games, you would win 418 of them and lose 342.

With a 60% win rate, you would need to play 380 games, you would win 228 of them and lose 152.

With a 65% win rate, you would need to play 253 games, you would win 164 of them and lose 89.

With a 70% win rate, you would need to play 190 games, you would win 133 of them and lose 57.

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.