Probability & Dice

[Sorrowdusk]

I was wondering, how do you calculate probability of multiple dice rolls?

I'm not sure how the math works.

Say for example you're using a percentile system and something has "potential" to happen on a roll of exactly a 1 on a d100. So it has a 1% chance.


Assuming you actually roll a 1, then there is still only a 10% chance of confirming (say by rolling a 1d10 and getting 1).

What's the actually over %chance of making both rolls successfully and how do you calculate that?

 

[Cadence]

I think the easiest way to visualize this problem is a tree diagram. A nice explanation is at: http://www.mathsisfun.com/data/probability-tree-diagrams.html

Google also has a variety of videos on the subject. One that seemed particularly reasonable is at:

[ame="http://www.youtube.com/watch?v=PVF5QBMF4lk"]http://www.youtube.com/watch?v=PVF5QBMF4lk[/ame]


The tree diagram for your particular problem would first branch into ten values for the first die, 0 to 9 (just like the example on the web-link branches into two values - head and tail for the first flip). Each of those then branches into ten values of 0 to 9 (like the second column of heads and tails for the second flip). That gives you 100 end branches, each with a probablity of 0.01 (since each one has 1/10 x 1/10 when you multiply them out...just like each branching in the web-page example above is 0.5x0.5=0.25). Now you wouldn't want to draw out all of those branches, but you can still count up how many of them would be exactly a one on the first roll and a one on the second roll (only one end branching out of the 100 possible ones). So now you add up the probabilities of all of the end-points you are worried about (just the 1 and 1 in this case) and get 0.01=1% (in one example on the web-page they wanted the chance of "at least one head", which went with three values of 0.25, so 0.75=75% total) . A tree diagram that had three sets of branches (instead of two like this example) would help you figure out the chances of getting each possible ability score on 3d6 (like the first example in the video, but values of 1 to 6 instead of just H and T).

The more mathy ways of calculating probabilities like these are usually the same as what the tree diagram represents graphically, they just assign symbols to everything in sight and use the "multiplication rule", "conditional probabilities", the "addition rule", and "combinatorics".

 

[Li Shenron]

I was wondering, how do you calculate probability of multiple dice rolls?

Simply by multiplication:

P { first roll succeeds AND second roll succeeds } = P { first roll succeeds } x P { second roll succeeds }

Which in your case would be:

P { 1% AND 10% } = P { 1% } x P { 10% } = 0.01 x 0.1 = 0.001 = 0.1%

The combined probability is the multiplication of the two individual probabilities only when the two "events" are indipendent, which is the case of rolling two dice separately.

 

[delericho]

I was wondering, how do you calculate probability of multiple dice rolls?

I'm not sure how the math works.

Say for example you're using a percentile system and something has "potential" to happen on a roll of exactly a 1 on a d100. So it has a 1% chance.


Assuming you actually roll a 1, then there is still only a 10% chance of confirming (say by rolling a 1d10 and getting 1).

What's the actually over %chance of making both rolls successfully and how do you calculate that?

For unrelated rolls, it's a simple matter of multiplication. So, that first roll has a 0.01 probability of success, the second has a 0.1 probability, and so the overall total is 0.01 x 0.1 = 0.001 probability (or 0.1%).

Likewise, the odds of rolling 18 on 3d6 are 1/6 x 1/6 x 1/6, or 1/216.

However, it gets a whole lot more complex when the dice are related. For example, the odds of rolling an 18 on 4d6-drop-lowest are not simply a matter of multiplication. That's where those tree diagrams that Cadence mentioned come in very handy.

However, it's also worth noting that very often you're not interested in a single outcome, but a spread of results. For example, what is the most likely outcome of rolling stats using the 4d6-drop-lowest method (that is, what 'value' of point buy does it approximate, and how much deviation can you expect). Unfortunately, you don't have to go at all far before probabilities become hard!

 

[ggroy]

Easiest way to do things systematically, is just write a computer program which enumerates all the dice possibilities in a similar manner to above mentioned tree diagrams.