Image copyright AFP Image caption Members of the Brazilian Frigate Constituicao recovering debris in June 2009

Statisticians helped locate an Air France plane in 2011 which was missing for two years. Could mathematical techniques inspired by an 18th Century Presbyterian minister be used to locate the mysterious disappearance of Malaysia Airlines Flight MH370?

In June 2009, Air France flight 447 went missing flying from Rio de Janeiro in Brazil to Paris, France.

Debris from the Airbus A330 was found floating on the surface of the Atlantic five days later, but the mystery of why the plane crashed could only be answered by finding the black box and the cockpit voice recorder.

You may think that having found the debris it would be easy to find the rest of the plane, but it's not that simple - after a number of days, the material would have moved with the ocean current.

Software does exist that can simulate how the debris has travelled from the initial impact. It is used regularly by the US Coast Guard.

But in this case, because this area near the equator is known for unpredictable currents - particularly at that time of year - it was no help.

Image copyright Getty Images Image caption Debris from the Air France crash is laid out for investigation in 2009

American, Brazilian and French ships, planes and submarines all searched for the plane, but they couldn't find it.

At this point France's aviation accident investigation authority, BEA, made a call to a group of statisticians in the US who had expertise in finding objects lost at sea.

Senior analyst Colleen Keller flew to France to help.

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"The French BEA had already done a wonderful job of coming up with different theories for why the aircraft might have crashed," she says.

They also had lots of data about historical crashes and the results of the searches that had already been carried out.

To turn all this information into numbers and probability, Keller and her team from Metron Inc in Virginia, relied on Bayesian statistics named after a British Presbyterian minister called Thomas Bayes.

This type of thinking allows you to assess various scenarios at once - even contradictory ones. The probability of each being true is brought together to give you the most likely solution. And if you find new information, you can revise your model easily.

Keller and her colleagues went through all the available information and assessed the uncertainties of each piece of data - applying Bayesian principles of probability to work out the most likely location of the plane.

The team split up the search area into a grid, and applied to each cell a figure representing the probability that the plane would be found there.

Image copyright AFP/Getty Images Image caption A Brazilian Air Force radar plane prepares to leave Fernando de Noronha airport to search for Flight 447

To calculate these figures, they first looked at the theories about what caused the plane to crash. For instance, they assessed the likeliness of various mechanical failures, and came up with a probability for each scenario.

They then assessed historical data from previous crashes, noting, for example, that planes were usually found very close to where they were last known to have been.

Finally, Keller and her team lowered the probability of the plane being found in locations that had already been searched.

"There are two components to Bayesian maths which make it unique. It allows you to consider all the data you have including the uncertainties which is very important because nothing is certain," says Keller.

"And to combine it all - it even allows you to combine views that contradict each other.

"For instance with the Malaysian search, you have that arc to the north and the arc to the south. It's either one or the other but it can't have gone both ways, but [Bayes] allows you to preserve all your theories and weight them."

The second benefit is that the Bayesian approach is very flexible, Keller says. It allows you to update your body of knowledge at any time. If something new comes up, you factor it in and update the probability map.

Image copyright EUMETSAT HO Image caption A 2009 infrared satellite image shows weather conditions off the Brazilian coast and the plane search area

In the case of the Air France plane, they could be sure that the plane had come down within a 40-mile radius of the last location pinged out by its on-board computer system.

Who was Thomas Bayes? Image copyright Other Born in London, 1702, the eldest of seven children

Studied logic and theology at University of Edinburgh from 1719 until 1722

Became the The Reverend Thomas Bayes, serving as minister in a Presbyterian chapel

But as a Nonconformist, he did not follow Church of England doctrines or practices

Best known for his mathematical work on probability, giving rise to Bayes' Theorem

Bayesian probability estimates are used all over the world, built into software that forecasts events including financial markets and weather

Died in Royal Tunbridge Wells in 1761

Yet this area was so huge that the investigators were struggling to know where to look.

The probability map Keller provided gave, by contrast, a much tighter area to search.

A team went out there, hoping that finally the mystery would be solved. But those hopes were dashed. There was no sign of the plane.

It seemed the statisticians could not help after all.

Some months later, though, Air France got back in touch and asked Keller to make one last attempt to analyse the data.

This time, she and her colleagues decided they were not happy with one of their initial assumptions.

The historical data showed that after a crash, the black box will be emitting a signal in 90% of cases.

In the immediate aftermath of the crash, search teams had spent a lot of time sweeping the areas close to the last known location, listening for the ping of the black box or voice recorder.

They had heard nothing. So Keller and her team had decided there was a very low probability the plane would be found there.

But what if neither the black box nor the voice recorder were sending a signal?

Image copyright Getty Images Image caption Alain Bouillard of the BEA speaking about the Air France 447 black boxes, found in 2011

The Metron statisticians now adapted their model to this possible scenario and came up with a new area of highest probability.

A team returned to the scene to look - and this time they found the plane.

The mystery of the crash was solved. The black box and voice recorder data appear to show that the pilots were given faulty speed readings, responded inappropriately, and lost control of the plane.

It's very likely if we don't get any breakthroughs, [Malaysia Airlines flight MH370 is] at the bottom of the Indian Ocean and we will never find it, sadly Colleen Keller

"It still was a minor miracle that we found it," says Keller.

"It was lucky that the wreckage was on the bottom of the ocean floor, on a very sandy area. There were some areas down there that looked like the Himalayas - in terms of mountains, crags, and valleys."

If the plane had been in one of those areas, she says, "it could have been undetected forever".

Keller says she is not sure Malaysia Airlines Flight 370 will be found.

"It's a big world out there. And I know people are saying - how could you possibly hide or not find a Boeing 777?

"[But] it's very likely if we don't get any breakthroughs, it's at the bottom of the Indian Ocean and we will never find it, sadly."

Even finding debris might not mean finding the bulk of the plane.

"If we found wreckage at this point, it would tell us it was in one body of water rather than the other," Keller says. "But it's so long since the plane would have crashed that I don't think the wreckage is going to be very helpful."

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