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Zero-modified geometric dice problem

  1. Dec 7, 2011 #1

    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?!?
  2. jcsd
  3. Dec 8, 2011 #2

    Stephen Tashi

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    Science Advisor

    Do you get to re-roll with a particular die only if you hit with that particular die? Or do you get to re-roll with a particular die if you hit with some of the other dice in the previous roll?

    If you only get to re-roll with a particular die when you hit with that particular die, then you don't need to worry about the fact that there are n dice. You can find the expected number of hits for 1 die and mullitiply that by n.
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