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How many wins do I need to get from Bronze 4 to Diamond?

84 wins

With a 55% win rate, you would need to play 840 games, you would win 462 of them and lose 378.

With a 60% win rate, you would need to play 420 games, you would win 252 of them and lose 168.

With a 65% win rate, you would need to play 280 games, you would win 182 of them and lose 98.

With a 70% win rate, you would need to play 210 games, you would win 147 of them and lose 63.

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.