In order to justify today's stratospheric share prices, profits and productivity must grow at improbably high rates for an impossibly long time.

Though a town of just 16,000, Franschhoek is a significant culinary hub, with an improbably high density of excellent restaurants on its main street.

Her most notable physical feature is her blue hair, styled into an improbably high beehive.

That seems an improbably high estimate.

The little fence was improbably high.

In the United States, that ratio would translate into 15 million people, an improbably high figure, given latter-day apathy.

But some officials said they considered that an improbably high figure.

The odds are stacked improbably high against us.

He said, though, that Mr. Gross's estimate of 1,600 dead was improbably high.

Feir felt himself passing through an improbably high number of wards.