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Thread: Ah! Math! [Probability and Dice Masters]

  1. #1

    Ah! Math! [Probability and Dice Masters]

    [ all calculations are done with an appropriate program via http://anydice.com/ ]

    First, let's talk turn 1 and the probability associated with rolling and rerolling 4 sidekick dice for any type of energy, a specific kind of energy, and just sidekick fielding.

    http://anydice.com/program/5c9f

    Output 1 is for getting as much energy as possible, regardless of type, rerolling any sidekicks.

    Output 2 is for getting as much of a specific energy as possible, rerolling all other results except for the wild energy.

    Output 3 is for getting as many sidekicks fielded as possible, rerolling any energy.

    But what if you want a combination of choices? Say, one Bolt, one of either Shield or Mask, one Sidekick, and one that we really don't care about as long as it is an energy?

    http://anydice.com/program/5ca1

    How to read: I apologize for the tricky binary exponent setup I used to distinguish the different results when combined together, so I'll explain: each successful roll is given a value that differentiate it from a different roll or the sum of a different roll. Since the possible results are either success or failure, exponents of 2 will suffice for successful rolls and 0 will suffice for failed rolls. Add the success values together to get the corresponding output on the output table.

    So, a result of 1 means you got the Bolt but not any other successes, a result of 3 means you got the Bolt and either a Shield or Mask but not the Pawn or the other energy (more on this in a bit: just now caught this myself), a result of 7 means you got everything but the other energy, and a result of 15 means you got all successes.

    Now, about that thing I mentioned, since we're talking binaries here, a successful sidekick is a failed energy roll: in other words, that last program provides some false information and, quite frankly, makes it hard to read (and I admit, I'm typing this as I go).

    http://anydice.com/program/5ca3

    Now, with the above criteria (1 Bolt 1 Shield/Mask 1 Sidekick 1 Wild Energy) and this new program, a result of 4 is a success for both the Sidekick and the energy, while a result of 0 or 8 means only one succeeded (0 means two energy, 8 means two sidekicks). Thus, success for all four would be a sum result of 7, which has probability just over 17.5%.

    This brings me to a point I'd like to make regarding requiring a combination of dice: the more specific the result you're looking for is, the less likely you will get that result. This should be intuitive to anyone familiar with basic probability, but to give an example of the disparity between a more generic result and a more specific result Turn 1 in Dice Masters, look no further than the first program:

    http://anydice.com/program/5c9f

    Take a look at probability for a result of 4 on Outputs 1 and 3 (remember: Output 1 is for as much energy as possible, and Output 3 is for as many sidekicks as possible). See how much higher the probability for 4 energy on Output 1 is compared to the probability for 4 sidekicks on Output 3? That is one huge disparity to be honest (just to put it into perspective, you have a better chance of getting only energy for 10 turns in a row (provided you only roll 4 sidekick dice per turn) than you do of getting only sidekicks for just one turn.

    http://anydice.com/program/5ca8

    That doesn't mean you should build your team with only consistency in mind: if you did, the meta would most likely revolve around Power Bolt and Magic Missile and simply using sidekick dice to power all six out. This brings me to my next point: impact (or probability of the desired results affecting the game state) vs reliability (or probability of getting the desired results).

    It should come as no surprise that the most powerful teams have high impact and high reliability, and that the least powerful have low impact and low reliability. But, unless you just like math and probability as much as I do, finding the perfect balance between impact and reliability can be tricky.

    Using the product of impact and reliability to measure a build's power alongside linear programming, having equal impact and reliability is better than focusing strictly on one or the other. For example, if a build has 40% reliability and 40% impact, it is better in terms of power than a build with 10% reliability and 90% impact, despite having a lower sum of reliability and impact.

    Something I should note here is that "impact" largely depends on the meta in which you're playing, and, as such, is more of hypothetical probability than a theoretical probability: one build could have a high impact against one particular build and a low impact against everything else.

    That's it for now: let me know if this helped ya at all and any constructive criticism is welcome.

  2. #2
    Keep posting this kind of stuff. I love me some statistics. (I also am rusty as hell when it comes to the formulas. I should get back into it.)

  3. #3
    Quote Originally Posted by RJRETRO View Post
    Keep posting this kind of stuff. I love me some statistics. (I also am rusty as hell when it comes to the formulas. I should get back into it.)
    Thanks! Coming from a CCG background, I have to say the switch from a deck of cards to a bag of dice makes both setting up calculations and performing calculations so much more accessible: I like not having to worry about moving things from queues to arrays to stacks and other ridiculous feats.

  4. #4
    That doesn't mean you should build your team with only consistency in mind: if you did, the meta would most likely revolve around Power Bolt and Magic Missile and simply using sidekick dice to power all six out. This brings me to my next point: impact (or probability of the desired results affecting the game state) vs reliability (or probability of getting the desired results).
    Yes, the teams should be built for as consistency as possible while maintaining a solid win condition, but you need to buy things that will also do something, especially early. Polymorph has been a notable exception because of the enormous impact that it can have on a later turn.

  5. #5
    Could this be made into a regular blog perhaps?

  6. #6
    ...heck, I'd read it.

    Start posting in the blogs section, ELC!

  7. #7
    Quote Originally Posted by Dave View Post
    Yes, the teams should be built for as consistency as possible while maintaining a solid win condition, but you need to buy things that will also do something, especially early. Polymorph has been a notable exception because of the enormous impact that it can have on a later turn.
    Agreed: but just to be clear, I said (albeit briefly) that impact was more or less the probability of a desired result to impact the game state (which, I should clarify, I mean the current game state). This means that as a game progresses, a build will have variable impact.

    To give a generic Magic example, aggro has a higher impact early on than control, but control has a higher impact as the game progresses, especially once they stick their dedicated win condition.

    Actually ... you gave me an idea for the next post/blog in this series.

    Quote Originally Posted by Scorpion0x17 View Post
    Could this be made into a regular blog perhaps?
    Quote Originally Posted by RJRETRO View Post
    ...heck, I'd read it.

    Start posting in the blogs section, ELC!
    I was actually thinking about it, but I wanted to get community feedback on this first issue. Glad to see it was well received. ^_^

  8. #8
    Oh I wasn't using "impact" the way that you were, I just meant effect.

    Regardless, nicely done.

  9. #9
    Wow...this was heady and number-y, which are both good things. Those are good ways to stretch me and my own thinking.

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