From The X-Files to The Men Who Stare at Goats, the weird fringe of America's intelligence agencies – those top-secret departments that keep tabs on aliens and optimistically conduct warfare using parapsychology – exerts an enduring fascination in popular culture. Tales of soberly dressed men and women treating outlandish concepts with total gravity can be entertaining, to a point. But there's a risk, particularly in fiction, of over-egging the fantastical until it becomes tiresome.

In The Coincidence Engine, the first novel by journalist Sam Leith, the weird fringe in question is the DEI, or Directorate of the Extremely Improbable. Leith, it must be noted, does not approach his subject with utmost seriousness. "Our job," says Red Queen, the DEI's beleaguered head, "is to assess threats to national security that we don't know exist, using methods that we don't know work. This produces results that we generally can't recognise as results, and when we can recognise them as results, we don't know how to interpret them."

But even in a comic context, the risk of overdoing it persists, and the first half of The Coincidence Engine is a bit like someone telling you at length about a really zany dream they once had. A hurricane in Alabama assembles a 737 passenger jet out of junkyard waste with a male stripper at its helm. A DEI agent is paired up with a man who lacks the capacity to be surprised by anything. Which is just as well because a crazy mathematician in the French Pyrenees has unleashed on America a potentially devastating device that plays havoc with probability. Frogs fall from the sky. Cashpoints spew out dollar bills. All the narcoleptics in Mississippi fall asleep in unison while crossing roads.

At times, The Coincidence Engine reads like a lighthearted imitation of a Thomas Pynchon novel, mixing conspiracy paranoia with Americana and a dash of wilful silliness, but without the unhinged wit needed to make it all gel, nor the requisite deep focus on the subject matter. ("I don't know any maths," Leith confesses in his author's note – and it shows.)

Luckily, the book picks up as it goes along. This is because Leith eases up on the zaniness and lets his characters be (relatively) normal. At the centre of the frog storms and probability meltdowns is a hapless Cambridge maths PhD student called Alex Smart, who sets off on a road trip from Atlanta oblivious to all the hullabaloo. His destination is San Francisco, where he intends to propose to his girlfriend, and en route he frets about his relationship and marvels at everyday America, which is quite eccentric enough without the interference of coincidence machines and DEI agents.There is a good moment when a man with a gun tells Alex to put his hands where he can see them. The gunman realises he's trotted out a movie cliche just as Alex thinks to himself: "They say that. They actually do say that."

At the end, though, even the author seems to tire of his protagonist, who tends to feel sorry for himself. "I detest Alex, don't you?" the hitherto silent narrator interjects without warning. It feels like an apology, and elsewhere the text seems aware of its shortcomings, flagging up bad jokes and being consciously vague about complex physics. At one point a maths professor confesses that he once wrote a sci-fi novel "about someone who builds a sort of magic probability machine". It was, he says, "silly. Pure nonsense." It would be unfair to apply the same verdict to this novel – it rises above its silliness – but it's not a million miles off the mark.