# How do I calculate the probability of specific dice rolls?

My question is a bit simplified due to space constraints, but I'll expand.

Here is my problem: I'm trying to calculate the odds a specific combination of dice. If I roll a die 6 times, what odds of a specific set of rolls. That's an easy enough problem, but the complication is that the numbers I can get on each roll can vary.

For example:

On roll 1, I can get a 1 or a 2 On roll 2, I can get any number but 6 On roll 3, I can only get the number 1 On roll 4, I can get the numbers 1 - 3 Etc...

Now I've been doing this sort of manually. Multiplying the fractions together, to get the odds of it. But it seems like there should be a simpler way, and I just can't figure it out. I've got a bunch of combos to churn through, and will do it manually if I have to, but am trying to save some time!

Forgive my math ignorance, and thanks!

• No simplification can be made. Why do you think that since you have for every roll different conditions ? – callculus Jun 9 '17 at 21:01
• I just mean a simplified way of calculating it. – jagrakye Jun 9 '17 at 21:02
• You only can put together the denominators to $6^6$. And the numerators can be multiplied separately. – callculus Jun 9 '17 at 21:05
• Ok thanks I'll just work through it – jagrakye Jun 9 '17 at 21:06
• Do you have many sets of possible constraints (where a single set is, for example, "On roll 1, I can get a 1 or a 2 On roll 2, I can get any number but 6 On roll 3, I can only get the number 1 On roll 4, I can get the numbers 1 - 3 Etc..."? If it is just this single set then you can calculate the probability for each roll individually very easily (for your example, 2/6, 5/6, 1/6, 3/6, etc.) and multiply them since each roll is independent. – fractal1729 Jun 9 '17 at 21:08

The probability of getting any individual result from a set ${d_1,..,d_n}$ of results from one dice|roll is: $$\frac n6$$
Hence the probability of getting any individual result from a set ${d_{1},..,d_{n_i}}$ for the dice|roll $i$ from 1 to N dices|rolls is: $$\prod_{i=1}^N\frac {n_i}6$$
For example, the probability of getting 1 from dice|roll 1 and 2,3,4 or 5 from dice|roll 2 is ${4 \over 36}$