Image copyright Thinkstock

It's difficult to keep a conspiracy under wraps, scientists say, because sooner or later, one of the conspirators will blow its cover.

A study has examined how long alleged conspiracies could "survive" before being revealed - deliberately or unwittingly - to the public at large.

Dr David Grimes, from Oxford University, devised an equation to express this, and then applied it to four famous collusions.

The work appears in Plos One journal.

The equation developed by Dr Grimes, a post-doctoral physicist at Oxford, relied upon three factors: the number of conspirators involved, the amount of time that has passed, and the intrinsic probability of a conspiracy failing.

He then applied his equation to four famous conspiracy theories: The belief that the Moon landing was faked, the belief that climate change is a fraud, the belief that vaccines cause autism, and the belief that pharmaceutical companies have suppressed a cure for cancer.

Dr Grimes's analysis suggests that if these four conspiracies were real, most are very likely to have been revealed as such by now.

Specifically, the Moon landing "hoax" would have been revealed in 3.7 years, the climate change "fraud" in 3.7 to 26.8 years, the vaccine-autism "conspiracy" in 3.2 to 34.8 years, and the cancer "conspiracy" in 3.2 years.

"The mathematical methods used in this paper were broadly similar to the mathematics I have used before in my academic research on radiation physics," Dr Grimes said.

Building the equation

To derive his equation, Dr Grimes began with the Poisson distribution, a common statistical tool that measures the probability of a particular event occurring over a certain amount of time.

Using a handful of assumptions, combined with mathematical deduction, Dr Grimes produced a general, but incomplete, formula.

Specifically, he was missing a good estimate for the intrinsic probability of a conspiracy failing. To determine this, Dr Grimes analysed data from three genuine collusions.

The first was the surveillance program conducted by the US National Security Agency (NSA), known as PRISM. This programme involved, at most, 36,000 people and was famously revealed by Edward Snowden after about six years.

Image copyright Reuters Image caption Dr Grimes analysed genuine cases of collusion, such as the PRISM surveillance programme, to come up with his estimates

The second was the Tuskegee syphilis experiment, in which the cure for syphilis (penicillin) was purposefully withheld from African-American patients.

The experiment may have involved up to 6,700 people, and Dr Peter Buxtun blew the whistle after about 25 years.

The third was an FBI scandal in which it was revealed by Dr Frederic Whitehurst that the agency's forensic analysis was unscientific and misleading, resulting in the imprisonment and execution of innocent people.

Dr Grimes estimates that a maximum of 500 people could have been involved and that it took about six years for the scandal to be exposed.

The equation he created represents a "best case scenario" for conspirators - that is, it optimistically assumes that conspirators are good at keeping secrets and that there are no external investigations at play.

Connecting the dots

Crunching the numbers from the three known conspiracies, Dr Grimes calculated that the intrinsic probability of a conspiracy failing is four in one million.

Though this number is low, the chance that a conspiracy is revealed becomes quite large as time passes and the number of conspirators grows.

The Moon landing hoax, for instance, began in 1965 and would have involved about 411,000 Nasa employees. With these parameters, Dr Grimes's equation suggests that the hoax would have been revealed after 3.7 years.

Image copyright NASA Image caption The Moon landings are the subject of a well-known conspiracy theory

Additionally, since the Moon landing hoax is now more than 50 years old, Dr Grimes's equation predicts that, at most, only 251 conspirators could have been involved.

Thus, it is more reasonable to believe that the Moon landing was real.

Prof Monty McGovern, a mathematician at the University of Washington, said the study's methods "strike me as reasonable and the probabilities computed quite plausible".

Dr Grimes added: "While I think it's difficult to impossible to sway those with a conviction... I would hope this paper is useful to those more in the middle ground who might wonder whether scientists could perpetuate a hoax or not."