# Thread: Dice Roll Game Probabilities

1. ## Dice Roll Game Probabilities

Hello everyone, I have a bit of a probability question for you as it has been quite awhile since I have been in school, and I can’t remember the equations for figuring out probabilities.
I am developing a dice rolling game where-in players roll 1 to 3 6-sided dice of a particular color vs. another player. On one end, you have an attacker and on the other a defender. There are 5 different types of dice; Red, Orange, Yellow, Green, and Blue. When rolling, an attacking player will have so many ‘hits’ negated by the opposing player’s ‘blocks’. Each dice has a set amount of ‘hit’ sides and ‘block’ sides as listed below:
Red Dice= 5 Hit Sides, 1 Block Side (or think of it like 2-6 are hits, 1 is a block)
Orange Dice= 4 Hit Sides, 2 Block Sides (or think of it like 3-6 are hits, 1-2 is a block)
Yellow= 3 Hit Sides, 3 Block Sides (or think of it like 4-6 are hits, 1-3 is a block)
Green= 2 Hit Sides, 4 Block Sides (or think of it like 5-6 are hits, 1-4 is a block)
Blue= 1 Hit Side, 5 Block Sides (or think of it like 6 is a hit, 1-5 is a block)
As should be apparent, certain dice are better on offense and certain dice are better on defense. As an example, An attacking player rolls 3 Red dice, a defender counters with 3 Blue dice. The attacker rolls 3 hits, the defender rolls 2 blocks. The 2 blocks rolled by the defender negate 2 of the attackers hits; so when the final it tallied, the defender takes 1 hit.
A second example, The attacker then throws 2 Orange dice, and rolls 1 hit. The defender rolls 3 green and gets 2 blocks, negating the attackers hit. The defender gets 0 hits on him/her.
The problem I am having is figuring out the probability of certain rolls vs other rolls. Is there a formula that I could use to determine the probability of getting 1 hit, 2 hits, and 3 hits respectively for each type of dice vs the other dice? IE: 1 Red Dice vs 1 Blue dice would 'hit' _ _% of the time.
I hope I am being descriptive enough. Below is a link to an Excel spreadsheet that is showing what I am trying to accomplish. I think this may clear it up better.
https://dl.dropboxusercontent.com/u/...blilities.xlsx
If anyone can help, it would be very, very much appreciated! Thank you for your time and have a great day!

2. ## Re: Dice Roll Game Probabilities

Simply use the Binomial pmf. The no. of hits taken seemingly is

You just need to use independence property to list the joint pmf table of , and can calculate the required pmf accordingly.

3. ## Re: Dice Roll Game Probabilities

Forgive me for sounding like a complete idiot, but its been over a decade since my last stat and prob class. Is there any way you could elaborate on that or post a sample of one equation?

4. ## Re: Dice Roll Game Probabilities

OK I try to do the first example for you - Attackers w/ 3 Reds vs Defenders w/ 3 Blues.

For each red dice, the probability of obtaining a hit in a roll is and therefore the total number of hits in 3 independent red dice follow , i.e. having the pmf

Similarly the distribution of the number of blocks is also

By independence the joint pmf is just the product,

Finally let to be the number of hits. You may form the following table for the transformation:

So this summarize how to map the 16 support points of to . To calculate the pmf of one just need to sum the corresponding entries.

E.g.

and just use the joint pmf listed above to calculate it.

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