PokéBase - Pokémon Q&A
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I would also like to know exactly how throwing mud at the Pokémon affects the flee rate, and how throwing bait does.

Not again :/
what do you mean by not again?

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Each Pokémon in the Safari Zone has a set catch rate and an escape rate.
Throwing Bait lowers the escape chance by 1 stage, but it also has a 90% chance of lowering the catch rate by 1 stage.
Throwing Mud, however, has the opposite effect; it makes the catch rate higher by 1 stage and the escape chance higher by 1 stage.
These stages are similar to Pokémon stat boosts. 0 is the starting stage and it can be raised to 6 stages or -6 stages.
According to Bulbapedia:

At the end of each turn, a random number between 0 and 255 is generated. If it is less than the modified stat stage, the Pokémon escapes.

This determines whether it escapes or not.
I hope this helped!

Source: Johto Safari Zone

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