Probability of a result from 3d6, lowest to highest

Question

I have a fascination with tabletop sports games, and through that, I've developed an interest in probability. That said, it's not my strong suit, so I wanted to pose this question because I think involves a few different probability principles to solve it.

Here's the premise...the game uses a roll of 3d6 to determine the result of a play. The dice are read from lowest to highest, as in:

a roll of 3, 6, 2 would be 2-3-6 a roll of 1, 4, 1 would be 1-1-4 etc.

That roll is then looked up on a chart to determine the result.

I was curious about the probability of different results coming up so that if I wanted to make a house rule, I would know which result would be the best place to modify.

So, what I do know is that for probability, you multiply chances together, correct? So without the low to high rule I discussed above, any number would have a 1/6 * 1/6 * 1/6 chance of occurring, correct?

How would I apply this principle to any result? My guess is something like:

1-1-4 = 1/6 * 1/6 * 5/6

only one chance for the first 2 die, and the last can be anything but 1, so 5 remaining numbers

2-3-6 = 1/6 * 4/6 * 4/6

first number can be 2 only = 1/6

second number can be any number but 2 or 6 = 4/6

second number can be any number but 2 or 3 = 4/6

Am I understanding this correctly?

Answer 1

Your answer's not correct. For 1-1-4, you have multiplied by $\frac56$, but it's not clear to me why. The correct calculation goes like this: there is a $\frac16$ probability of getting a 1 on the first die, a $\frac16$ probability of getting a 1 on the second die, and a $\frac16$ (not $\frac56$) probability of getting a 4 on the third die. This multiplies out to $\frac1{216}$. But there are three different orders in which the rolls could occur to give a result of 1-1-4, since 1-4-1 or 4-1-1 give the same result. So you must multiply the $\frac1{216}$ by 3, for a final result of $\frac1{72}$.

Similarly, the result for 2-3-6 is again $\frac1{216}$, but this time multiplied by 6 because there are 6 different orders in which the 2, 6, and 3 can appear; the final result is $6\cdot\frac1{216}=\frac1{36}$.

In general, the answer is:

• If the pattern you're looking up is XXX, with all three dice the same, the probability is $\frac1{216}$.
• If the pattern is XXY or XYY, the probability is $\frac1{72}$.
• If the pattern is XYZ, the probability is $\frac1{36}$.