How do the catch rates of Quick Balls and Timer Balls compare?

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We know Quick Balls are great early and Timers Balls are great later on. How do their catch rates change turn after turn? If there are any statistics about it, I’d like to see them. If it is different in different Generations, let’s say Gen 7 is the one I’d like to know.

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Timer ball

(1 + number of turns passed in battle * 1229/4096 (approx. 0.3))×, maximum 4× (13 turns)

Quick ball

5× if used on the first turn of a battle1× otherwise

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