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June 13th, 2012, 08:06 PM   #1
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Probability with dice and target numbers

Hi, this is my first time on the forums and I am having a very hard time with some math questions. I have tried searching the web for my particular question but most sites are concerned with the sums of 2 six-sides dice not target numbers. What I mean by target numbers is I am trying to figure out what the probability of rolling "at least" one 1, 2, 3, etc. on multiple dice. The reason I am doing this is because I am trying to build an excel sheet to help me make better decisions with a table top game I play called "Heavy Gear Blitz". I very fun game, everyone should check it out.

One of the facets of the game is when you roll a double/triple/etc. six instead of getting a six, you get six plus how many other sixes you have rolled. For example, on four dice I rolled a 6, 6, 6, 4. I would then have the final score of 8 because of my first six and then +1 for every six after it. This really puts the math a bit beyond me. I have gotten a bit of the target number math done, but I am really not sure if I have it right or not.

This is what I do:

3 six-sides dice being rolled
chances of not rolling at least a 6 on the first die = (5/6)
chances of not rolling at least a 6 on the second die = (5/6)
chances of not rolling at least a 6 on the third die = (5/6)
I then multiple all the results together = .5787 or 57.87%
I want to find out the result of rolling a 6 so I then subtract the number I found from 1 = .4213 or 42.13%

I hope this is all right. I am not really sure how to do the math for what I was talking about early with "at least" rolling a "7". Thank you for your time and any help you can give!
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June 13th, 2012, 11:24 PM   #2
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Re: Probability with dice and target numbers

You are correct. The chance of rolling at least one six is 1-(5/6).
Now, you want to know the chance of rolling n dice and getting exactly k sixes.

Suppose you roll 8 dice, and you are interested in getting 3 sixes.
To get the sixes on (for example) the 3rd, 6th, and 8th rolls, the probability is

and this probability is the same for any choice of three dice out of the eight.
But there are ways to choose those three dice.
These are independent roll results, not overlapping, which means we can add up the probabilities.
So the probability of exactly 3 sixes out of 8 dice, for a score of 6+1+1 = 8, is , about 10.42 %.

In general, for n dice and exactly k sixes, the probability is .
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