The political fringes are dominated by conspiracy theories that far too often encroach on the usually more reasonable political middle. In this post I want to look at some math behind grand conspiracies, which involve many people keeping a secret, as contrasted with small conspiracies, which only require a small, tight group to maintain.
One classic example of an alleged grand conspiracy is the supposedly-faked Apollo moon landings of the late 1960s and early 1970s. More recent examples of alleged conspiracies that touch on politics include the birth certificate of President Obama and some of the opposition to anthropogenic (human-caused) climate change. In short, the math on which alleged grand conspiracies depend usually falls apart quickly. In contrast, I will attempt to demonstrate in an upcoming post that money laundering and sexual indiscretions, as two examples, are often a case a small conspiracy repeated, especially because applicable laws are weak and rarely enforced (a probabilistic clue).
Rock bands and rocky relationships
In almost any conspiracy, somebody is telling a lie and at least one other person knows it. And here is where the math come in, because each relationship in the conspiracy requires “trust against the lie.” When the trust breaks in one fiber of the web, the lie cannot stand.
Indulge me here enough to stretch to an example about the rock and roll bands of my youth in order to demonstrate some math. I was a member of one in my teen years, and the close intertwined relationships, plus the probability that one of the band members is much more talented than the others, virtually insures high drama at some point, and we went through a couple of very messy “personnel changes.”
In a simple folk-rock duo like Simon and Garfunkel, there is only one relationship and that one strained to the breaking point after their top-selling Bridge Over Troubled Water album, with Paul Simon going on to greater success than Art Garfunkel (who I really like, by the way). In the Beatles, with four members during their heyday, the number of two-way relationships rises to 6, which is 3! (or factorial – 3 times 2 times 1).  At various points, every one of those six relationships fractured, so that by 1969, the group collapsed.
Fleetwood Mac first formed as an all-British blues band, then broke up and re-formed several times in the 1960s and 1970s before its commercially successful five-person line-up, which raises the number of two-way relationships to 24 (4!). Add the sexual tension and you get decades of high drama, break-up, “tell-all books” and reformation. By the time you get to a group of seven members, like the first successful lineup of Chicago Transit Authority (later just Chicago), you are at 840 one-on-one relationships, and you need to find a “normal” way to rotate people into and out of the band, which they have usually done, surviving today with (I think) four original members.
You will get “this guy”
The same math applies to trying to keep a conspiracy together. As a somewhat-silly example of the collapse of “relationship-trust” math, rally organizers at a recent event in Montana featuring Donald Trump tried to “stuff the crowd” behind him with enthusiastic supporters, and instead got this guy who “broke the trust” and made incredulous faces for several minutes before getting ejected: 
My point here is that attempted grand conspiracies are plagued with huge numbers of relationships, each one requiring “trust against the lie” in order to hold up. Because of that, the odds of success quickly become tiny, which should cause any person understanding the math to pause at the assertion of any large-scale conspiracy. You will eventually get “this guy” who has a secret to tell. 
And conversely, if you don’t get this guy, then you likely do not have a conspiracy. Given the thousands of people involved in the Apollo space program, and hundreds involved in the most intimate details of that project, the pure probability of every one of these relationships, had there been a lie to hold up, is down very close to zero. And even then, it has been well demonstrated by several sources that the technologies required to create the purported fake landings were either not available in 1968 or would have been harder to pull off than the moon landing itself. 
The same math applies the Obama birth certificate controversy, although perhaps with smaller, yet very sufficient numbers of trust relationships. The complications here are time and precognition. Any purported conspiracy of birth has to go back to 1961 and held since then, with the precognition by all parties in that relationship that this baby was destined to become very important in adulthood. That the myth persists in a large subset of the population is a whole separate subject of collective delusion which I have covered in a prior post. Collective (or mass) delusions are, in contrast, probabilistically quite easy to pull off.
As for anthropogenic climate change, this is a case of “probabilities upon probabilities,” which is both the strength of the argument and a source of much of the political misdirection. I discussed one of these probabilities, the “sand pile effect,” in an earlier post about “rain events.” Scientists rarely give a “yes/no” answer on extreme weather events and their causation, instead talking about these phenomena in probabilistic terms. That frustrates the innumerate public and feeds conspiracy theories, but, as this blog often points out, nature itself is one huge mass of probabilistic events going back for you to the very time of your conception.
However, climate scientists have hit the issue from hundreds of different, usually independent sources of evidence, from ancient ice cores to historical weather data to tree rings to satellite mapping, and many, many more sources. When most of them probabilistically converge on the same answer from different directions, the odds that we are deep into anthropogenic climate change get higher, and the odds of conspiracy, already tiny, go well below the point of measurability.
When you know the math, then you begin to recognize allegations of grand conspiracies for the political misdirection that feeds them. To quote Neil DeGrasse Tyson on the alleged conspiracies involving science, “The good thing about science is that it’s true whether or not you believe in it.”
- If you need to count them: Paul-John, Paul-George, Paul-Ringo, John-George, John-Ringo, George-Ringo. I’ll let you debate over which one was the most talented. The general formula here, given n participants, is (n – 1)! combinations of one-to-one relationships.
- The video is here.
- Weiner, Sophie. “Why Faking the Moon Landing Was Impossible.” Popular Mechanics, 14 Nov. 2017.