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I'm trying to come up with a probability for a game I play with a friend of mine. In the game, units "attack" by rolling six-sided dice; either 2 or 4 sides of the die count as a "hit" when rolled, depending on certain circumstances. The specific situation I am trying to figure out the probability for is:

you get n dice to roll.

assume the probability to hit is "p"

for each die, you can re-roll if you miss on your first roll.

for each die, you keep re-rolling as long as you hit.

what is the probability of k hits?

the thing that is causing me problems is the fact that you can reroll the die if you miss at first. So, instead of each die being a geometric random variable, each die is a zero-modified geometric random variable with Pr(no hits) = (1-p)^2.

If the die were simple geometric R.V.s, their sum would be a negative binomial R.V. which I could easily evaluate. Does a similar property hold for zero-modified geometrics R.V.'s? If so, what is the parametrization? If not, how could I figure out this probability?

One idea I had: let Z = the total # of hits,

Pr(Z=k) = Ʃ Pr(X of n dice roll one or more hits) * Pr(Z = k | X dice have one or more hits, n - X dice miss totally).

I got bogged down in the details of this and wasn't able to come up with something that worked; any ideas?!?

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# Zero-modified geometric dice problem

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