Does Le Pen have a chance of winning French presidency?

The opinion polls say no, and conventional wisdom says she hasn't a hope.

Centrist Emmanuel Macron is a long way ahead in the polls - so is there any way far-right leader Marine Le Pen could still pull off a shock victory in France's presidential election on 7 May?

What do the numbers say?

How the first round was won

Since Mr Macron's impressive first-round victory, the polls suggest he is around 19 points ahead of his rival. The margin has narrowed but it is still wide.

The omens look good for him too. He won 8,657,326 votes in the first round on 23 April, almost a million more than his National Front runner-up.

And many voters who backed other candidates first time around say they plan to back him rather than Ms Le Pen.

He has strong backing from key political figures: outgoing Socialist President François Hollande and candidate Benoît Hamon on the left and defeated Republican candidate François Fillon and ex-President Nicolas Sarkozy on the right.

Marine Le Pen's big coup has come from attracting ex-candidate Nicolas Dupont-Aignan, who won 1.7 million votes and has been promised the role of prime minister.

So what could go wrong for Emmanuel Macron? Abstention is the biggest threat, says political scientist Aurélien Preud'homme.

"If there is a very weak turnout for Macron and very strong support for Le Pen, she could go above 50% of the vote," he told the BBC.

So could apathy win Le Pen the vote?

Marine Le Pen does not even need to go above 50% in the polls to become president, according to one expert, as long as Macron voters stay away in big enough numbers.

All she needs is to advance a little more in the polls, and this is how she could do it.

"If she gets 42% of the vote, which isn't impossible, and Macron gets 58%, normally she loses the election," physicist and Sciences Po political expert Serge Galam told RMC radio.

"But if 90% of people who said they would vote for Le Pen do it, and at the same time only 65% of people who declared they would vote for Macron actually do it, then it's Marine Le Pen who wins the election with a score of 50.07%."

It is not that the polls are wrong, it is just that they cannot gauge the level of voter apathy in advance. Under Serge Galam's mathematical formula, he gives three examples of how Marine Le Pen can win, where she is candidate "A turnout x" and Emmanuel Macron is "B turnout y" with a Turnout (T).

He calls this model "differentiated abstention".

Whether or not the polls might fail, it is worth pointing out that almost all the polling organisations were uncannily accurate in predicting where the top six candidates would finish.

Marine Le Pen's campaign has hit the ground running

US pollster Nate Silver gives Marine Le Pen no chance at all, arguing she is in a far worse hole than Donald Trump ever was.

• The issues dividing Le Pen and Macron

• The final candidates

• Five things the election has told us

• Why the election matters so much

But Ian Bremmer of risk consultancy Eurasia Group puts the chance of a Le Pen victory at 30%.

"I feel he's going to win but it's not a safe bet. It's significantly about turnout and there are externalities that are very plausible."

One of those external factors, he believes, might be a terrorist attack on a greater scale than the murder of policeman Xavier Jugelé in central Paris three days before the first round. Or perhaps an outbreak of fake news.

What are the chances of a low turnout?

It could happen, as the 7 May run-off comes in the middle of a holiday weekend. Why make a special effort to stay home and vote on Sunday when Monday is a public holiday? It could work both ways but the Macron vote is more city-based and more likely to venture away from home.

"Neither Le Pen, nor Macron" is the message from several groups of French voters

Then there are the significant groups who cannot bring themselves to back an economic and social liberal in Emmanuel Macron. Many are attracted to the hashtag #SansMoile7mai (Without me on 7 May) or describe themselves as "Ni Le Pen Ni Macron" - neither one nor the other.

Another group is Sens Commun (Common Sense), a socially conservative group opposed to same-sex marriage and adoption. Their leader sees the "political decay" of Mr Macron as no different to the "chaos" offered by Le Pen.

More than four out of 10 voters do not appear convinced by either candidate, and that could translate into a high rate of abstention.

Historically, however, turnout in the second round is as high as 80%, so there would need to be a dramatic decline. Latest polls suggest three-quarters of voters intend to cast a ballot.

Can Le Pen improve her chances?

Her main hope rests in rallying undecided voters, both those planning to cast a ballot and those who are not.

The far-right leader's second-round focus has been to stage an energetic campaign that targets her rival as the "candidate of finance" and paints her as the patriotic choice.

• Macron heckled by pro-Le Pen workers

She has distanced herself from her own party, suspending her leadership of the FN until after the election.

Most dramatically, she upstaged Mr Macron by making a surprise appearance with striking factory workers at Amiens while he was talking behind closed doors to their union leaders.

Voters were unimpressed by Mr Macron's decision to celebrate in posh La Rotonde bistro

By contrast Mr Macron initially appeared flat-footed, beginning with a celebration party on Sunday night at a pricey Paris bistro and what sounded like a rebuke from President François Hollande. "We need to be extremely serious and mobilised and not think it's a done deal, because a vote is earned, it's fought for."

For Marine Le Pen a lot will hang on the big TV duel on 3 May. In earlier TV debates Mr Macron performed noticeably better.

Can she attract new voters?

Marine Le Pen has already achieved a record number of votes for her party and is even trying to distance herself from the brand by restyling herself as the people's candidate with ex-banker Emmanuel Macron as a "candidate of the oligarchy".

• In depth: Meteoric rise of Emmanuel Macron

• In depth: Is Marine Le Pen far right?

The key lies with two blocs: the right wing of François Fillon's Republican support base and the radical left who opted for the surprise package of the first round, Jean-Luc Mélenchon.

Together the two candidates polled almost 40% of the vote.

Mr Mélenchon initially refused to back either run-off candidate, raising the possibility that a section of anti-globalisation voters could find common cause with Marine Le Pen.

However, stung by criticism, the far-left leader insisted he had not adopted a "neither-nor" stance and warned that a vote for the National Front would be a "terrible mistake".

Ms Le Pen is heading for Nice, where more than 80 people were killed in a lorry attack last July

Political scientist Aurélien Preud'homme says many on the right are leaning towards Marine Le Pen and it just depends on whether she can shift the focus during the final days of campaigning to her strongest suit, security and terrorism.

"What stops the right backing Le Pen is Europe and the euro," says Mr Preud'homme. Most Fillon voters are of a certain age and they're the ones who worry about leaving the euro - it would be very worrying for them."

If Trump can do it, why not Le Pen?

Marine Le Pen's father Jean-Marie Le Pen has already said she should have adopted a more aggressive campaign style in the manner of Donald Trump.

He overturned a double-digit lead over Democratic candidate Hillary Clinton to win the US election last November.

But she has little time left and may not find enough voters to win over.

It is worth remembering that when she won the first round of elections in six of France's 13 regions in 2015, she looked on course to change the course of French politics. But it never happened and her party was defeated in the second round.