1. Probability Function Help

During my college days, I used my statistics classes to help me understand the probabilities of various roleplaying games, including Vampire, the Masquerade. In that game, players build a number of 10-sided dice into dice pools, and roll them to accomplish various actions in the game. The Storyteller assigns each action a difficulty between 4 and 10, and each die that equals or exceeds that number is counted as a success. Each number greater than 1 but less than the difficulty is a failure, but each 1 that is rolled on the dice cancels a success. If no failures are rolled, but one or more 1s are, the failure is considered a botch, with spectacularly bad results.

Like I said, years ago, I created a function to model the probability of rolling s successes with a given dice pool of p, with a difficulty of d.

Within the past couple of years, though, the publishing company released the 20th anniversary of the game, with a few minor rule tweaks. Unfortunately, I haven't had a course in statistics in a good long time, and can't come up with a way to tweak my function to reflect the updated rules.

In the 20th anniversary edition, a player can assign a specialty to certain skills or rolls. In those circumstances, a roll of 10 counts as two successes on the roll, instead of just one.

So while for non-specialty rolling, my function still works just fine, it doesn't capture the "double counting" that goes on when characters are making rolls in their specialties. How can I adjust my function to reflect this rule change?

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