What's the name for this statistical fallacy? I was told an anecdote by someone today who was trying to prove a point regarding safety. They said "50 people have been living in [area a] for the past two years one year (apparently I remembered the conversation wrong) and there have been no incidents, therefore the area is safe for more people to live there."
[area a] happens to be what the government considers a high-risk zone, with an elevated threat to personal safety (specifically death). I know this person's reasoning is flawed but I'd like to know the exact name and explanation of that flaw, because I feel this is quite a common one.
I see two main factors contributing to the error:


*

*Small sample size

*The risk is heavily weighted on the "death" side of things, it's not an elevated risk of getting a paper cut


How would I call out this flaw despite the fact that the person is technically right in saying "There have been no incidents"?
Edit for clarity: This [area a] is equivalent to a building, and is occupied by more than just this sample set. The area is within a larger region in which there is an elevated risk of harm or death, and the area offers no special protection against it. Incidents of risk are rare, but certainly higher than the background rate and do occur in this larger region.
 A: It also sounds like the parable of the thanksgiving turkey: 
http://www.businessinsider.com/nassim-talebs-black-swan-thanksgiving-turkey-2014-11
Every morning the farmer feeds the turkey well.  After 1000 days the turkey argues that the farmer is benevolent and the pattern will continue.  But day 1001 is Thanksgiving...
(Note for global readers: Thanksgiving is a US holiday on which it's customary to eat turkey.)
A: This is not a fallacy, but rather the Problem of induction, as popularized by David Hume.
A: General case of survivors fallacy:
Looking only at/for things that didn't fail skews your perception. This may lead you into an untested and thus failure intolerant behaviour.
The usual example is observing planes returning from air combat:
"Do you need to increase armor in places where the returning planes were hit?" Supposedly it's where planes are likely to be hit.
However the answer is counter-intuitively "No, because that's where planes are likely to be hit and survive." So hits there are survivable anyway.
You achieve real results, when you increase armor in places the "survivors" have not been hit, because that's where the "non-survivors" were hit.
For your case (singular):
Under the precondition of moving a single person into an area with incidents leading to deaths.
Do I need to move into a sub-area that has not been hit by an incident?
No, for those sub-areas you simply have no conclusive data.
Instead you need to move into a sub-area where incidents do happen but don't lead to deaths. The goal is not to have no incident but to survive it, in case it happens, right?
If you don't want the incident to happen, you shouldn't move into the larger area in the first place!
For your case (plural):
If you want to move a statistically relevant number of people into the area where incidents are survivable, you need to first check, if the reason incidents are survivable is low population density in said area.
If incidents are survivable in low density population areas, moving people in wouldn't make the people safe but the area unsafe.
Another view on things:
If there are 1000 people in the larger area, of which 20 died in the last incident, then there are still 980 survivors left to tell the tale. Is it safe, because more people survived than died?
Surely most of the 980 people weren't even close to the 20 that died, when it happened. Does it become any safer, if you just ask those?
Can you ask the 20 dead people, if they'd still consider it safe?
Bottom line is, you'll feel safe as long as you ask survivors, who didn't witness the incident. Since you can only ask survivors, it's probable they didn't witness the incident.
Hence, Survivors fallacy.
Related fallacies:
Others have mentioned other fallacies. I don't want to repeat them in detail. However I do see that they apply as well. So here's a compilation and the aspects why they apply and why they are different:


*

*Survivors fallacy: Concentrating on favourable results only.

*Texas Sharpshooter fallacy: Choosing a sub-sample in hindsight.

*Hot hand fallacy: Interpreting random variation of results as indication of probability distribution, especially when looking at most recent history.

*Small numbers law: Relying on insufficient data.

*Base rate fallacy: Underestimating the importance of general information in favour of more specific information.


There's another well-known fallacy that I originally mistook for "Hot hand". Now that I think about it, it actually doesn't apply:


*

*Gambler's fallacy: Misunderstanding the law of large numbers to mean that independent events would even out in the long run.


It's kind of inverted Hot hand fallacy:
Falling for "Hot hand" you'd bet on what happened most often in recent history, because it seems more likely.
Falling for "Gambler" you'd bet against what happened most often, because the opposite seems in need to even out in the long run.
A: This sounds like the hot hand fallacy to me. 
https://en.wikipedia.org/wiki/Hot-hand_fallacy
When teaching intro stats I found a lot of students fell for this fallacy. So the idea is in basketball sense, he made X amount of shots he is more likely to make the X + 1 shot. Same idea here X amount of people live here with no incidents therefore no incidents should occur if X + 1 people are present. 
A: This is the base rate fallacy:

If presented with related base rate information (i.e. generic, general information) and specific information (information only pertaining to a certain case), the mind tends to ignore the former and focus on the latter.

In this case, the base rate of death is quite high, but the specific information is that there are at least 50 people living in the area who have been unharmed.
A: I don't have a specific name for the fallacy, but here is a reference that I think is relevant (along the law of small numbers line):
The Most Dangerous Equation
Also a statistical rule of thumb (see section 2.9) says that an approximate 95% confidence interval for the 2 year incidence rate given none in 2 years would be from 0 to $\frac{3}{50}$, so the incidence could be as high as 6%.  So if you moved another 1,000 people in then it would not be surprising to see 60 incidences in the next 2 years.
Thinking about it more, if the small area was chosen because of no incidences and there are some in the larger area, then this would be a variation on the Texas Sharpshooter Fallacy.
A: The plural of "anecdote" is not "data."
(Also quoted at https://stats.stackexchange.com/a/8404.)
A: Statistical inference becomes invalid when there is no variability  -and in this case, the variability is non-existent. So the only way that the argument:

"50 people have been living in [area a] for the past two years and
  there have been no incidents, therefore the area is safe for more
  people to live there."

can be examined, is non-statistical, i.e. deterministic. Therefore the argument is methodologically valid (not factually correct) only if it is read as

"50 people have been living in [area a] for the past two years and
  there have been no incidents, therefore the incident rate in the area is and will remain zero."

Wow. I am impressed with the confidence level of the person saying this.
Any implied inference of the type "if the rate is zero in the sample, we expect it to be "small/acceptable/"normal" in the population" (which is how one could understand the "it is safe to live there" assertion) is garbage, both because there is no base to extrapolate from sample to population, but also because there is no base to extrapolate from past/present to the future.
As Fisher would say, "get more data".
