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Astrology can help most tests, but has a chance of actually taking away a success from the roll, thought it is unlikely.

Ignoring the fiction and focusing only on the rules, is there any situation where accepting the astrology helping die actually reduces the chance of succeeding at a roll?

The factors one can play around with are: number of dice rolled, shade of the dice, obstacle, whether the roll is open or not. (Supposedly using deeds artha might also factor in. Feel free to consider it, also, if you feel like it.)

My intuition suggests that it would have to be a situation where one has a good probability of exactly matching the obstacle and little chance of missing it by only 1 success. However, achieving these at the same time seems difficult or maybe impossible. Rolling one white die with obstacle 1 does gain benefit from the astrology die, for example.

I am well aware that there are other reasons for not always helping everything with astrology; this is not the issue at hand. Let us focus on the probabilities, please.

I will accept an answer that gives a concrete case where the probability of failure with astrology is greater than without astrology, or a correct mathematical proof that such can never be the case. Extensive numerical calculations covering most possible cases might be accepted, also.

Please note that positive expected value of the astrology die does not constitute a proof.

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Thanuir, is there any particular answer that you feel is the most helpful? I personally lean toward Angelo or stormsweeper. – user2102 May 25 '12 at 12:16
@JoshuaDrake I did not learn anything new from either, but did find them less unhelpful than others. – Thanuir May 25 '12 at 14:29
up vote 5 down vote accepted

Consider the situation of rolling 10D against Ob1

You have about 99.9% chance of getting 1 success or more. You have about a 98.9% chance of getting 2 successes or more.

If you add an astrology die, you get a 50% chance of improving your chance of success to 100% (ie not needing any successes from your 10D). This gives you a 99.95% chance of beating Ob1.

OTOH, you also get a 16.7% chance of needing 2 successes to get Ob 1. This reduces your chance of beating Ob1 to 99.78% - less than the original chance of 99.9%

NB values from Scott Gray's dice pool calculator - assumes 6s don't explode

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If those two cases are factored together, is the final percentage the same, lower, or higher than the 99.9% without the Astrology die? – SevenSidedDie Sep 27 '12 at 18:06
It's LOWER. The calculation is 99.9*(2/6) + 100*(3/6) + 98.9*(1/6). That's a 1/3 chance of the astrology die rolling 2 or 3 (no effect), a 1/2 chance of rolling a 4, 5, or 6 (succeess), and a 1/6 chance of rolling a 1 (need two successes). The total, after all of it is added up, is 99.78%. So in that case, it's a bad idea to roll the astrology die. – Paul Marshall Sep 28 '12 at 17:54
Note that the chance of the astrology die taking away a success is only 1/12, not 1/6. Even considering that the example works, assuming that the astrology die can take away no more than 1 success (this is unclear in the rules, at least to my reading). This all supposing I did not make any mistakes. – Thanuir Jun 14 '13 at 14:11

I don't believe so, no. The average contribution from the astrology die is positive. It grants a success on 4,5,6 and only takes one away on a 1, so its expected contribution is +1/3. This isn't quite as good as a regular black die (1/2) but it's much better than no die at all (0)!

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Positive expected value does not, by itself, imply that the probability of getting a given result increases. For a silly example, suppose you roll d100: if the result is 100, +1000 successes; otherwise, -1 success. The expectation is positive, but most of the time that die would not be helpful. – Thanuir Apr 17 '12 at 5:49
@Daenyth the question is about an absolute case, not the average or even majority cases. – user2102 May 21 '12 at 16:49
@Daenyth you seem to read the question the opposite of me, or possibly I am reading this answer incorrectly. The OP's comment on this answer may explain why he has not accepted it. – user2102 May 21 '12 at 17:10
@Thanuir: In that light it may be a better option for Mathematics or Cross Validated, because at that point the only relation to RPG gaming is the fact that it's using dice. It's a statistics question at its core. – Daenyth May 22 '12 at 13:12
@Daenyth Well, no, because the question is about the impact of the rest of the game system, not strictly about the roll of the dice in a simplified case. That's what this answer misses: it talks only about the normal case, which is not what the question is asking. It's asking does there exist a case (regardless of how unusual it is) where the die's expected contribution is negative instead of positive? – SevenSidedDie May 22 '12 at 17:34

It depends. If you grant your players (/your dm grants you) guaranteed successes then yes.

Your chance of having success increases with every added dice (but does not reach 1), so adding dice is always increasing your chance of success. With the one exception that if you already guaranteed reached your target adding a possibility to reduce the number of successes (because in this situation your chance of success would be 1 without the dice, and < 1 with it).

As far as I understood burning wheel, there is no situation where one could have a guaranteed success by default so as long as you didn't house-ruled that otherwise you chances are always better with more dice as long as the possibility of that added dice to add a success is larger that the possibility of that dice to take a success.

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The astrology helping die may take away a success, so the situation is not obvious. – Thanuir May 23 '12 at 13:51
@Thanuir It still is. The probability of it helping is significant larger than it hurting (with the stated exception) – Angelo Fuchs May 24 '12 at 6:22
If the situation is obvious, then presenting a formal proof is trivial. But consider a situation where the chance of getting enough successes is almost 1, say 1-e for some small number e. Should you still accept the astrology die? Can this happen in play? How small does e have to be in order for this to happen? – Thanuir May 24 '12 at 7:50
@Thanuir I see what you mean, my math days are too long ago so I cannot give a formal proof. I will keep the answer up still for the special situation of guaranteed successes (although they are not in the default construct of this game) because as far as I can tell this is the only situation where e is small enough (As far as I can tell this is only true if e = 0) – Angelo Fuchs May 24 '12 at 19:31

I'm not sure anyone could easily post a proof here - you're looking at some wonky math with binomial distributions. Complicated even further because the Astronomy die is open-ended in both directions.

Your example of W1 is easy, though - the chance of failure against Ob 1 is exactly the same with or without the Astrology die.

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I know the calculation is fairly nasty - by ignoring some open-endedness I can get fairly nice estimates, but since I had to ignore the open-endedness, I achieved neither necessary nor sufficient bounds. That was all back-of-the-envelope calculations, though. – Thanuir May 23 '12 at 13:50

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