This feels like the wrong conclusion to draw (especially the second part), but I suck at statistics so someone please explain where I'm off.
Extreme example: suppose a company has 100 employees. Four of them are in the union bargaining team and they just happen to all work for the widgets department because of an accident of history or randomness etc. The widgets department has 10 people overall.
Now the company decides for business reasons entirely unrelated to unions, that thanks to GPT they don't need so many people in the widgets department. So they fire 8 people from widgets. Assume further that they went out of their way to protect the union bargainers: of those 8 people fired, only 2 are in the union bargaining team. (That means that the remaining 2 widgets people are both union bargainers.)
Even though the company went out of their way to protect the union bargainers, statistically they only laid off 8% of the company, but 50% of union bargainers. Would that be illegal or even unfair?
(In the real world, I am fairly sure that union bargainers are concentrated in some departments. Mostly because it attracts certain kinds of people, and different departments also attract certain kinds of people.
But, of course, I have no clue what the reality inside of Bandcamp was like. I'm just speculating about hypothetical examples.)
My example crucially hinges on the company being able to convincingly argue that they singled out that particular widget department for reasons unrelated to the union. If they can't make that argument, or someone even manages to proof that they fired from the widget department _because_ of the union people, then they would be in deep trouble.
P(lose job | bargainer): 50%
P(lose job | not bargainer): 6.25%
-----widgets team-----
P(lose job | bargainer) 50%
P(lose job | not bargainer) 100%
-----everyone else----
P(lose job | bargainer) NaN
P(lose job | not bargainer) 0%
----------------------
This isn't the cleanest example, since the comparison at the bottom can't be drawn at all. But the way the problem is narrated, it would counterfactually have exhibited the paradox.
Edit: also, another confounding factor that hasn't been discussed is job role/department. Layoffs and union membership are definitely not evenly distributed across role/department, but there is not enough info in the article to know if the layoffs were targeting specific roles or uniform across the company.
Modelling any of these assumptions yields an arbitrary probability of your own choosing.
>> You might want to ask yourself "Do I really believe that, if I thought I might be about to get fired, that would have no influence on whether I tried to join a union?"
You actually want the odds of a specific set of 40 individuals getting laid off. It looks like Bandcamp had 118 employees and 60 layoffs per various news outlets (and on the wikipedia page [0]). Assuming each employee was equally likely to get laid off given the target retention [1], you could compute this from the binomial distribution using "number of ways to predetermine those 40 are getting laid off, plus 20 from the remainder, divided by number of ways to select 60 individuals out of 118".
Or, more concisely, (78 choose 20) / (118 choose 60), which is actually closer to 10^-16 (2 orders of magnitude more unlikely than the simple approximation of 2^-40, which assumes an infinitely large pool).
[0] https://en.wikipedia.org/wiki/Bandcamp : "Songtradr stated that only 60 of Bandcamp's previously 118 workers had been offered a contract."
[1] This is by no means guaranteed.
Well, close. I've computed the odds that at least 40 of a group of 67 special people get laid off.
There is another comment in this thread suggesting that 67 is the number of bargaining team members. I don't know, because I haven't read the article. So my calculation might or might not be right, depending on whether you interpreted the article correctly or the handful of people taking the other position did. My interpretation was that 40 out of 67 union members got fired, which appears to have been wrong.
However...
> You actually want the odds of a specific set of 40 individuals getting laid off.
I definitely don't want that. You never want to make a comparison against a specific outcome when you ask "what are the odds?" like this. All specific outcomes are rare, so that question will never tell you anything informative. (I almost wrote "will never tell you anything useful", but if what you're looking for is a scapegoat, you might find the calculation you propose useful for that. It's not useful for anything else, and frankly it's a disgrace that you suggested it.)
edit:
This is what the article says:
> “On Monday, October 16, 2023 over half of Bandcamp was laid off as a result of Epic Games’ divestiture to Songtradr,” Bandcamp United said in a statement. “Of those laid off, 40 were in the union bargaining unit out of a total 67 members. None of the eight (8) democratically elected bargaining team members received a job offer.”
So it looks like there are these groups:
- bandcamp employees (number unknown)
- union members (number unknown)
- union members on the bargaining team (67)
- union members on the bargaining team who got laid off (40)
- union members elected to the bargaining team (8)
- union members elected to the bargaining team, who got laid off (8)
This suggests that the calculation I gave was the same one that was sought, and also that the base rate, as far as we believe in it, was over 50%. Remember that shifting the odds of being fired from 50% to 53% more than doubled the odds of seeing the pattern we did see.
118 (ironically, not 120) per https://finance.yahoo.com/news/half-bandcamps-120-person-sta...
I incorrectly conflated the number of laid off union members with the total number of "special" individuals, while only 8 of them were actually "special" (i.e. democratically elected bargaining team members). It'd be more accurate to describe that probability of the impact to the bargaining team as (110 choose 52) / (118 choose 60) which comes out to about 0.35%, not too far off from the 2^-8 estimate (0.4%) that your approach of using the binomial distribution on this same set would result in. Both approaches would yield similar results, and the result is still somewhat in the realm of plausible deniability.
> All specific outcomes are rare, so that question will never tell you anything informative.
I'm not describing any specific outcomes in this case, though. You could throw up your hands and say "oh, we laid off these specific 60 people, there's only a 10^-34 chance of picking exactly those 60 people out of 118". That would absolutely be useless.
I'm describing the event that "all of this one group got laid off" as a subset of the space of possible outcomes given what we observed, and ascribing a probability to that event.
I will admit I used the wrong inputs, but I stand by my approach.
Given that 50% of all employees were laid off, what were the odds of at least 40 employees out of 67 randomly-selected employees getting laid off?
[This is the LibreOffice formula: "in 67 trials which each have a 0.5 chance of success, what is the probability of seeing between 40 and 67 successes?"]One-tailed p-value of 0.07, two-tailed p-value of 0.14. So the odds are about 1 in 7, or 1 in 14.
You might want to ask yourself "Do I really believe that, if I thought I might be about to get fired, that would have no influence on whether I tried to join a union?" [Answering this question in the negative means you can justify using the one-tailed p-value, but it also means the odds of getting fired should be substantially above 50%.]
Note also that this number shifts substantially as you move the odds of getting laid off away from 50%. At 53%, you have a one-tailed value of 16.44%. At 47%, it's 2.5%.