How are a dice pool's probabilities affected by being allowed/forced to add dice then drop high/low dice?

Question

I'm working on a homebrew system. The base mechanic, without getting too involved on specifics, is roll a pool of d6s (1–5) and compare each vs. a target number (Generally 4–6); any that are equal to or higher count as a success and the more successes you get the better.

That is easy to calculate. What I'm having trouble with is how another proposed mechanic will affect the probabilities. If you are allowed/forced to add some (one or more) extra dice to the pool and ignore the same number of lowest results (for beneficial circumstances) or highest results (for detrimental circumstances), how will that affect the likelihood of getting a certain number of successes?

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An example for clarification: You would roll to accomplish something and the appropriate stat has 3 dice. Every die that comes up 4+ is a success. 1 success is more a technical success that comes with downsides, 2 successes is a decent success, 3 is success and then some, etc. This aspect is already working as intended. The mechanic I would be introducing is either helping or hindering a roll by adding a die and dropping either the lowest or the highest respectively. E.g., an ally helps the character, so the 3d6 now becomes a 4d6 with the lowest die result being ignored.

Answer 1

Yes this is easy to calculate

The original dice pool (ODP) of n dice, with each having a probability p of success is a binomial distribution. The probability of exactly k successes is:

$$ f(k;n,p) = Pr(X=k)=\binom{n}{k}p^k(1-p)^{n-k} $$

Add a die and remove the highest

Adding a dice and removing the highest gives us the following cases to deal with:

1. The ODP has 0 successes. The additional die will have no effect: if it is a success it must be the highest and is therefore removed, if it is a failure then all of the dice are failures (including the highest one that gets removed).
2. The ODP has k(>0) successes. If the additional die is a failure, one success will be removed, if it is a success, it will add one success and one success (from the highest die) will be removed for no net effect.

Ignoring the unusual case of 0 successes in the ODP, the die has a 1-p chance of reducing successes by 1.

Add a die and remove the lowest

Adding a dice and removing the lowest is the reverse of the previous:

1. The ODP has n successes. The additional die will have no effect: if it is a failure it must be the lowest and is therefore removed, if it is a success then all of the dice are successes (including the lowest that gets removed).
2. The ODP has k (<n) successes. If the additional die is a success, one failure will be removed, if it is a failure, it will add one failure and one failure (from the lowest die) will be removed for no net effect.

Ignoring the unusual case of n successes in the ODP, the die has a p chance of increasing successes by 1.