I'm having trouble figuring out how to determine the probabilities for this specific dice mechanic (the game in question is Bang, the Dice Game but I'll try to generalize it).
In this game, rolls act like Yahtzee, where you roll three times, choosing any of them to keep or reroll based on what you get. However, one of the results (lets say 1) has an IMMEDIATE negative consequence when you roll it (losing 1 health for example). You can reroll this result to a better one, but that roll is still counted.
My question is, for a Yahtzee-like game, what is the probability of getting this negative result (a 1 in 6 chance) a given number of times, including those obtained during intermediate rolls? For example, if you roll five dice and get two 1s, then reroll all five and get three 1s, then reroll all five and get no more 1s, that's a total of 5.