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

80 wins

With a 55% win rate, you would need to play 800 games, you would win 440 of them and lose 360.

With a 60% win rate, you would need to play 400 games, you would win 240 of them and lose 160.

With a 65% win rate, you would need to play 267 games, you would win 174 of them and lose 93.

With a 70% win rate, you would need to play 200 games, you would win 140 of them and lose 60.

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.