The more people who are in on a conspiracy, the more likely it is that someone will spill the beans. The logic is clear, and far from original, and now someone has done the maths: Most famous alleged conspiracies are simply too large to be true.

Dr. David Grimes of Oxford University works in cancer research and is also a science writer, drawing plenty of messages from those with a theory to push.

“A number of conspiracy theories revolve around science,” Grimes explained in a statement. “While believing the moon landings were faked may not be harmful, believing misinformation about vaccines can be fatal. However, not every belief in a conspiracy is necessarily wrong – for example, the Snowden revelations confirmed some theories about the activities of the U.S. National Security Agency.

“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,” Grimes continued. “To do that, I looked at the vital requirement for a viable conspiracy – secrecy.”

Grimes has published a formula in PLOS ONE that takes into account the number of people who know the secret, the length of time they have to stay silent and the likelihood of death sealing their mouths forever.

Crucially, Grimes needed to know how likely it was that any individual would let something slip. This may vary with circumstances, so Grimes investigated three scandals where word got out: the NSA's Prism project, the Tuskegee syphilis experiment and FBI forensic abuse.

“For a conspiracy of even only a few thousand actors, intrinsic failure would arise within decades,” Grimes writes in the paper. Grimes estimated that it would have taken 411,000 people to fake the Moon landings and 405,000 to cover up if climate change was a fraud. If pharmaceutical companies have an effective cure for cancer but are covering it up, they would need to silence 714,000 people, many of whom would have to have been willing to watch loved ones die without breaking ranks.

Covering up a vaccine-autism link would have involved 22,000 people at the World Health Organization and the Centers for Disease Control and Prevention, but the number blows past 700,000 if all medical researchers working on vaccines are assumed to be in the know.

None of the conspiracies could last four years, Grimes calculated, let alone decades, unless those involved are far more naturally secretive than FBI and NSA staff.

For a plan to last just five years, no more than 2,521 people can be involved, and that drops to 1,000 if you want it to last a decade, Grimes said.

Grimes ruefully noted: “This will of course not convince everyone; there's ample evidence that belief in conspiracy is often ideological rather than rational.” Nevertheless he adds, “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.”