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Here is the Kleptomancy rune description :

PASSIVE: After using an ability, your next basic attack within 10 seconds has 25 Range icon bonus range. If the attack is against a champion, you gain Gold 5 Gold. Occasionally, you will gain a consumable item instead. If you gain a consumable item while your inventory is full, the consumable will instead be consumed.

What are the exact odds to receive the consumable? Does it change during the game?

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    There was nothing revealed by developers. It's possible to calculate after thousands of matches, but would ultimately never be 100% confirmed. There is anecdotal evidence that it prioritizes items that benefit you (eg: grant you magic elixirs more often than AD ones if you're AP)
    – Oak
    Nov 15, 2017 at 9:19

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There's a post on reddit regarding this. Though it is just an assumption, TS has done great work on calculating all the chances. From what I can see, the chances do scale up.

According to the graphs and tables presented in that discussion we can track that that overall proc rate increases from ~30% at the game start to the max of ~45% proc chance in a game that lasted for more than 52 mins. That is linear proc chance we can see in the graph. We can still observe spikes up to 70% in the mid to late game and down to less than 20% in the early to mid game.

Here I will cite the author: Looking at the data points, there are a couple things we can assume: - The chance of Kleptomancy activating scales up with game time. - Getting sustain potions scales down with game time. - Getting combat potions, Control Wards, and Farsight Wards scales up with game time. - Getting Minion Dematerializers, Sight Wards, and Potion of Rouges appear to stay relatively the same over the course of the game.

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