Major conspiracy theories such as a faked Moon landing cannot be true or they would have been exposed within a few years a scientist has concluded.
Dr David Grimes, an Oxford University physicist, worked out a mathematical formula to calculate the chances of a plot being leaked by a whistle-blower or accidentally uncovered.
He was able to show that the more people involved in a conspiracy, the shorter its lifespan is likely to be.
For a plot to last five years, the maximum number of plotters turned out to be 2,521. To keep a scheme undetected for more than a decade, fewer than 1,000 people could be involved, while a century-long deception had to include fewer than 125 collaborators.
Since Neil Armstrong became the first man to walk on the Moon in 1969, conspiracy theorists have proposed that the landing was faked. They point to photographs showing the American flag blowing in the breeze, despite there being no atmosphere and the fact the astronauts' shadows do not match the lunar module light source.
But applying Dr Grimes' formula to the Moon landings, which would have involved an estimated 411,000 people who worked at Nasa, any hoax would have been found out in three years and eight months.
"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," said Dr Grimes. "To do that, I looked at the vital requirement for a viable conspiracy - secrecy." Dr Grimes's conspiracy equation factored in conspirator numbers, length of time, and even the effects of conspirators dying, whether of old age or non-natural causes.
Also required was a realistic estimation of any one individual disclosing a conspiracy. This was based on three genuine conspiracies, including the United States' National Security Agency Prism spying programme revealed by whistleblower Edward Snowden.
Dr Grimes, whose research appears in the online journal Public Library of Science ONE, said his work was inspired by the numerous communications he receives from people who believe in science-related conspiracies.
"A number of conspiracy theories revolve around science," he said. "While believing the Moon landings were faked may not be harmful, believing misinformation about vaccines can be fatal. 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.
"If we are to address the multitudinous difficulties facing us as a species, from climate change to geopolitics, then we need to embrace reality over ideologically motivated fictions. To this end, we need to better understand how and why some ideas are entrenched and persistent among certain groups despite the evidence, and how we might counteract this."
Four famous conspiracy theories and the length of time it would take before their secrets leaked out into the public domain:
Many people still believe that Nasa faked its Apollo missions, and has never landed a man on the Moon.
However, Nasa had 411,000 employees at the time of the Moon landing and one of them should have blown the whistle within three years and eight months.
Cancer cure cover-up
It has been claimed that big pharmaceutical companies are hiding a cure for cancer to maintain profits for chemotherapy drugs.
But if everyone from Novartis, Pfizer, Roche, Sanofi, Merck and Co, and Johnson and Johnson, to GlaxoSmithKline and AstraZeneca were involved it would amount to 714,000 people and the story would get out within three years and three months.
Climate change fraud
If man-made climate change was a myth, a huge number of scientists, academics and policymakers would need to be involved in the cover up.
Including members of the American Geo-Physical Union, American Academy for Advancement of Science, European Physical Society, the Royal Society, and Nasa, as well as published climate scientists, 405,000 would have to keep quiet. That would be possible only for three years and nine months.
Unsafe vaccine conspiracy
If unsafe vaccines were being produced, a cabal of the Centre for Disease Control the World Health Organisation, and almost certainly pharmaceutical companies would be needed adding up to 736,000 people. Such a plot would crumble within three years and two months.