A physicist has calculated the probability that four widely believed conspiracy theories could have lasted this long without being uncovered. And - mathematically speaking - it's not looking good for Moon-landing deniers or anti-vaccers.

The 2016 study revealed that for an old-fashioned cover-up to stay under-wraps for a decade, fewer than 1,000 people would need to be involved.

And to remain secret for a century, the number of people 'in on it' would need to be below 125. Considering that the most popular conspiracy theories would realistically involve thousands of people, the odds aren't great.

"My results suggest that any conspiracy with over a few hundred people rapidly collapses, and big science conspiracies would not be sustainable," physicist David Grimes from the University of Oxford in the UK told The Guardian at the time.

Although Grimes's research usually focusses on cancer, he also regularly writes science stories for the media, and was inspired to conduct this study after being bombarded with emails from people believing in popular conspiracies.

"It is common to dismiss conspiracy theories and their proponents out of hand but I wanted to take the opposite approach, to see how these conspiracies might be possible," he explained. "To do that, I looked at the vital requirement for a viable conspiracy – secrecy."

He applied his equation to four big conspiracy theories: the belief that vaccines cause autism; that climate change isn't real; that the Moon landing was a hoax; and that a cure for cancer already exists, but is being hidden by big pharmaceutical companies.

Taking into account the number of people that would need to be involved to pull off each of these four scenarios, and how long they've been around, his results suggest:

  • If the Moon landing was really a hoax, around 411,000 NASA employees would have known, and it would have been revealed in just 3.7 years.
  • A cover-up of the link between vaccines and autism would have been exposed anywhere between 3.2 and 34.8 years after it started.
  • A climate change 'fraud' would have been revealed in 3.7 to 26.8 years.
  • The suppression of a cancer cure would have been leaked by someone inside big pharma in just 3.2 years.

The calculations were based upon the best-case scenario for conspirators, which means that Grimes's equation assumed that conspirators are really good at keeping secrets, and that no external investigations were going on.

To create the equation, Grimes had to first establish the probability of a cover-up being exposed over time. To do this, he considered three IRL, confirmed conspiracies, including the US National Security Agency (NSA) surveillance program that was famously uncovered by Edward Snowden after just six years.

He also factored in the Tuskegee syphilis experiment, where the cure for syphilis - penicillin - was purposefully withheld from African-American patients. That conspiracy involved up to an estimated 6,700 people, and was eventually revealed around 25 years later by one of the doctors involved. 

Grimes also analysed the FBI forensic analysis scandal. All of these cases are testament to the fact that conspiracies and mass cover-ups do occur, but that they don't last very long.

"Not everyone who believes in a conspiracy is unreasonable or unthinking. I hope that by showing how eye-wateringly unlikely some alleged conspiracies are, some people will reconsider their anti-science beliefs," explained Grimes. 

"While believing the Moon landings were faked may not be harmful, believing misinformation about vaccines can be fatal."

The results were published in PLOS One. 

A version of this article was first published in January 2016.

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