D&D 5E Rolled character stats higher than point buy?

[delericho]

That's the Gambler's Fallacy right there!

The Gambler's Fallacy only applies when the rolls are independent of one another. Because you only reroll some results, that's not the case here.

 

[FrogReaver]

Dang it's hard to break up such long posts on a cell phone. Many things I'd respond to in here but it's too difficult without being able to easily break them apart.

We seem to be talking past each other, so permit me to clarify my fundamental point.

Imagine that we know are using 4d6k3 six times to generate a set of six ability scores, but we don't know whether or not we can re-roll sets that don't meet a certain criteria. In fact, one of us is allowed to re-roll, but the other is not, but we don't know this! In fact, it will be decided by a flip of the coin which of us would be allowed to re-roll, and this coin will be flipped after we have each generated our set of six.

We roll our stats as instructed. We expect the average distribution to be about 16/14/13/12/10/9, because that is what is expected for 4d6k3. As it turns out, neither of our sets would qualify to be re-rolled anyway.

At what point is one of us expected to have a higher set than the other, based on a re-roll that was never used?

Explain to me how the one that was allowed to re-roll (but never did) has an expectation of a higher stat distribution than is expected for 4d6k3!

See, this is the reality of rolling for scores. Sets of six are not generated from some average that is observed after the effect. Time only moves one way, and so does causality. Your possible decision to discard a set doesn't make any rolled set higher than it was before! If you discard one unsuitable set after another, five, ten, fifty times until you finally roll a set that doesn't qualify to be discarded and re-rolled, that final set has exactly the same expectation as every other: 16/14/13/12/10/9 in our case.

Re-rolling does not increase any set you roll, discarded or not. This is the Gambler's Fallacy; the idea that if the last seven spins of the roulette wheel have spun a red number, that the next spin is more likely to be black, on the grounds that eight reds in a row only has a 1 in 256 chance. This is bogus! If you're interested, the chances of red/black remain equal, because the sequence '7 reds followed by 1 black' also has only a 1 in 256 chance.

With (possible) discarding, the sets you discard have absolutely no effect on the next set you generate. Discarding doesn't give a mathematical expectation of generating higher scores.

The fact that you can observe averages of already generated sets and change the average of those by discarding low sets doesn't actually increase any rolled set at all! You increased the average, not by increasing any scores, but by taking low results away! This manipulation of the observed average has no impact on character creation beyond taking low scores away! It doesn't increase any kept set at all!

 

[Arial Black]

The Gambler's Fallacy only applies when the rolls are independent of one another. Because you only reroll some results, that's not the case here.

Thank you! At last!

If you discard a rolled set, then the next set you roll is entirely independent of the next set you roll! That previous set has no effect whatsoever on this set; it doesn't make this set of rolls higher than they would've been without a previous set being discarded.

So, yes, this is the real, bona fide Gambler's Fallacy in all its glory!

 

[Maxperson]

Thank you! At last!

If you discard a rolled set, then the next set you roll is entirely independent of the next set you roll! That previous set has no effect whatsoever on this set; it doesn't make this set of rolls higher than they would've been without a previous set being discarded.

So, yes, this is the real, bona fide Gambler's Fallacy in all its glory!

Which isn't the point at all. So what if they are independent. If you are only keeping the sets that are well above average, the set you end up with will always be much higher than the average set. The average is meaningless.

 

[FormerlyHemlock]

<nt>

 

[delericho]

Thank you! At last!

If you discard a rolled set, then the next set you roll is entirely independent of the next set you roll! That previous set has no effect whatsoever on this set; it doesn't make this set of rolls higher than they would've been without a previous set being discarded.

So, yes, this is the real, bona fide Gambler's Fallacy in all its glory!

You might want to re-read what I wrote, because you seem to have drawn the exact opposite conclusion from what it actually says. The second roll is not independent of the first - it depends on it for its very existence.

 

[EzekielRaiden]

You might want to re-read what I wrote, because you seem to have drawn the exact opposite conclusion from what it actually says. The second roll is not independent of the first - it depends on it for its very existence.

Which is why I did things like presenting the 4d5k3+3 as opposed to 4d6k3 reroll 1s. Because the d5-based set cannot possibly be the same as the unmodified d6 set--they're physically and numerically different generators, with no rejection rules involved. Yet the probability distribution for 4d5k3+3 is, in absolutely every sense, precisely identical to that of 4d6k3 reroll 1s. Every probability is identical, and there is no difference in the possible results that you can get.

Therefore, rigorously applying selection rules, and looking only at those things that come out after the selection rules have been applied, must be (in some cases) equivalent to changing the type of generator used, which *means* changing the distribution.

Or, another way to put it, [MENTION=6799649]Arial Black[/MENTION]: you seem to be focusing on P(x), where x is "a result that can be produced from rolling 4 six-sided dice and discarding the lowest of them." But none of us are focusing on that. We are focusing on P(x|C), where "x|C" is "result x as defined before, given that condition C has been met."

It is not the gambler's fallacy to assert that P(x) =/= P(x|C). It's Bayes' theorem. P(x|C) = P(C|x)*P(x)/P(C). These are not independent events. Whether a die, or set of dice, is rerolled does depend on the current state thereof.

 

[rballison]

The real reason that rolled stats are higher is that people can simply roll characters until they get one they like. The only limitation is their patience and time. The actual statistics don't really matter. Since you can discard unwanted results and start over, there is no downside to rolling repeatedly until you get a character that is good enough.

 

[FrogReaver]

Not the gamblers fallacy. Other than when you brought it up none of us have been talking about the intermediate probability of the reroll. We have all been talking about the probability of the final outcome given that rerolls exist. That's a much different scenario and it doesn't fit in the bounds of the gamblers fallacy.

Thank you! At last!

If you discard a rolled set, then the next set you roll is entirely independent of the next set you roll! That previous set has no effect whatsoever on this set; it doesn't make this set of rolls higher than they would've been without a previous set being discarded.

So, yes, this is the real, bona fide Gambler's Fallacy in all its glory!

 

[FrogReaver]

Gamblers fallacy. Given that the first coin flip was heads the next is very likely tobe tails.

Not gamblers fallacy: given that I can reflip a coin that lands on heads one time the probability of the final outcome being tails is 75%