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

4 wins

With a 55% win rate, you would need to play 40 games, you would win 22 of them and lose 18.

With a 60% win rate, you would need to play 20 games, you would win 12 of them and lose 8.

With a 65% win rate, you would need to play 13 games, you would win 8 of them and lose 5.

With a 70% win rate, you would need to play 10 games, you would win 7 of them and lose 3.

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.