"3 d6 Probabilties (for geniuses)" Topic

I am struggling with the calculations of probability for rolling 3 6 sided die and getting the following results:

i) 6 on any of the 3 die rolled.
ii) 5 or 6 on any of the 3 die rolled
iii) 4 ,5 or 6 on any of the 3 die rolled.

Can anyone help please?

There are 3 dice of 6 faces, 6*6*6 possible results = 216.
A unique result such as 6, 6, 6 would be 1/216
For one face on 6 the chance is of a 6 on 1 die and any other result on the other 2.
1/6 * 5/6 * 5/6 = 25/216
This can happen with either of the 3 dice so add up the 3 scores.
75/216 = 35%

No genius but…
i) 3/6 or 50% of 1 success.
ii) 6/6 or 100% of 1 success.
iii) 9/6 or 150% of 1 success.
hope this helps!

to re phrase to make it a bit clearer

i) 6 on at least one of the 3 die rolled. (includes rolling 2 sixes or 3 sixes)

ii) 5 or 6 on at least one of the 3 die rolled (includes rolling 2 sixes or 3 sixes or 2 fives or 3 fives)

iii) 4 ,5 or 6 on at least one of the 3 die rolled.(includes rolling 2 sixes or 3 sixes or 2 fives or 3 fives or 2 fours or 3 fours).

For one face on 6 the chance is of a 6 on 1 die and any other result on the other 2.
1/6 * 5/6 * 5/6 = 25/216
This can happen with either of the 3 dice so add up the 3 scores.
75/216 = 35%

But you also need to add:
The possibility of 6 on 2 dice and any other score on the other:
1/6 * 1/6 * 5/6 = 5/216
And this can also happen in 3 ways so:
15/216
Then there is also the possibility of 6 on on all 3 dice:
1/6 * 1/6 * 1/6 = 1/216
75 + 15 + 1 = 91
91/216 = 42%

A 5 or 6 on 1 die and any other result on the other 2.
2/6 * 4/6 * 4/6 = 32/216
This can happen with any of the 3 dice so add up the 3 scores.
96/216
Then there is the possibility of 5 or 6 on 2 dice and any other score on the other:
2/6 * 2/6 * 4/6 = 16/216
And this can also happen in 3 ways so:
48/216
Then there is also the possibility of 5 or 6 on on all 3 dice:
2/6 * 2/6 * 2/6 = 8/216
96 + 48 + 8 = 152
152/216 = 70%

A 4, 5 or 6 on 1 die and any other result on the other 2.
3/6 * 3/6 * 3/6 = 27/216
This can happen with any of the 3 dice so add up the 3 scores.
81/216
Then there is the possibility of 4, 5 or 6 on 2 dice and any other score on the other:
3/6 * 3/6 * 3/6 = 27/216
And this can also happen in 3 ways so:
81/216
Then there is also the possibility of 4, 5 or 6 on on all 3 dice:
3/6 * 3/6 * 3/6 = 27/216
81 + 81 + 27 = 189
189/216 = 87%

Good chat. Very interesting
may I chip in with some thoughts.

It may be simpler to calculate the probability of total failure . Take this away from 1 and what is left is the probability of success!

Chance of not gettinga 6 is 5/6. Chance of this happening on 3 dice is 5/6 x 5/6 x 5/6= 125/216

So if that is prob of failure. all othet outcomes woudl give 1 or more successes!

So prob is 91/216 of success.

A 5 or 6 would be 4/6x 4/6 x 4/6= 64/216 of failure. Thus 152/216 for success.

So in closing , find the chance it will not happen and take it way from 1.

Play around on this website: anydice.com

thanks for the solutions and especially how you calculated it! very useful.

There are 216 possible combinations of results. Out of those, 91 of the sets will contain a 6 on at least one die. Meaning that the probability of rolling one or more 6s is 91/216 or 42.13%.

For 5&6, P = 70.37%

For 4,5&6, P = 87.5%

Basically, confirmation of everyone else's answers.

I am with Martin. I find it easier to do the chance of it not occurring, and then subtract that from the total possible results.

If you're not afraid of a little math: look up binomial distributions. Those describe the number of x successes, on y tries, each with a z probability for success.

But with the low numbers of dice you need, it's indeed possible to compute it exhaustively as shown above.

If it helps, I found wakenney's answer the easiest to understand….

Other maths too much for my tiny brain…

John T

Edit: oh and thankfully I have more than one set of percentile dice!