Yesterday, I discussed how to estimate the odds for the Gamma World technology charts using the RandBetween function on an Excel spreadsheet.
But today, I want to look at an oddity in the charts themselves--or specifically at an oddity in Chart A. I have no idea if anyone else has remarked on this oddity. It may be quasi-common knowledge among some in the community or not. The only thing I can say is that I haven't seen it discussed, and I have been jumping around the blog posts on this issue a fair amount recently.
Here's the oddity:
Without explicitly computing the probabilities or doing random simulations, it would seem obvious that the longer you spend in sustained concentration trying to figure out an item, the greater your chance of success. The rules also state that while you can fiddle for as long as you want, any interruption means you must begin again at the start. It would seem, therefore, that you don't want to stop or be interrupted. However, while it may seem that way, it's actually not the case. For Chart A, at least, it's actually better to stop or be interrupted after just a few rolls (and thus begin again at the start) than it is to try to continue rolling.Now, it might be objected that this is a trivial observation, at least in certain cases. If you keep on getting high rolls, then you're probably moving closer to that skull and crossbones (again, henceforth A) and so in that kind of a case it's better to start over than to keep going. While this might be true, it's not precisely what I'm getting at. The falsity of the claim does not depend on the player knowing what his rolls are. The claim would be false even if the referee were rolling behind a screen.
Here's the basic point: while each roll gets you potentially closer to F, it also gets you potentially closer to A. However, it takes fewer rolls to get to F than to A. That implies that after a certain number of rolls (whatever they are) it might be better to start again at the beginning than to continue.
Remember the results of our first sample of 1000 attempts at rolling ten times:
F (success): 503
A (accident): 32
No Result: 465
Now let's look at what happens when we roll only five times:
F (success): 324
A (accident): 5
No Result: 671
The 5 accidents make sense, since we saw yesterday that we know for certain that after only five rolls there is only a 0.63% chance of getting to A.
So, while taking ten rolls as opposed to five increases the odds of success by 50% or so (503 as opposed to 324) , it also increases the chance of catastrophic failure by perhaps 600% (32 to 5). The chances are still small, but when it comes to, say, dying, I'd rather have a really small chance--0.63%--than a small chance--3.0%.
But by stopping at five rolls, don't you also sacrifice your chance of success? So, isn't there a trade-off between risk and reward?
No, actually, there isn't.
How about this strategy: Make five rolls. If you don't get to F (or A), stop, go back and try again from scratch.
The results of this strategy will roughly approximate those below:
F (success): 543
A (accident): 8
No Result: 449
(We get 543 by using this formula: 324 + (.324 * 671). We get 8 by using this formula: 5 + (.005 * 671).)
So, it's better to start again after five rolls, than to continue on to ten. It's better to be interrupted.
In choosing two sets of five rolls (if we need them), rather than ten, we have roughly the same chance of success (or even more of a chance according to one simulated set of 1000 iterations) but a much lower chance of catastrophic failure.
But, of course if we had been thinking clearly, we should have already had a hint of this phenomenon. To see this, consider the results of only four rolls:
F (success): 227
A (accident): 0
No Result: 773
We had 0 occurrences of A. Should we roll 1000 more times to see if this was a fluke? No. As we saw yesterday, it takes a minimum of 5 rolls to get to A. It is impossible to get to A in only four rolls. We should have known that. We could have known that by simply looking at the chart.
So, how about a strategy of choosing three sets of four rolls (if we need them)? Here are the results:
F (success): 539
A (accident): 0
No Result: 461
So, again, the chances of success are about the same, but now we have completely eliminated the chance for catastrophic failure.
Hurrah, we've just come up with a foolproof scheme for sussing out simple artifacts without risk of breakage, injury or death!
We've also shown how Chart A is slightly broken.
Does this sort of thing also apply to Charts B and C? I'll leave that for another time. For now, though, how can Chart A be fixed?
Three possibilities come to mind. One is to make the minimum path to A shorter than the minimum path to F. Another (though this is potentially far more lethal) is to give some chance of getting to A from any position or at least from more of them. Finally, we could simply decree that being interrupted or voluntarily stopping does not mean that you go back to S. Rather, you always start where you left off from.
We'll discuss these in another post.
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