LONDON — Ask for his opinion, and David Howe eagerly tells you how he hopes Britain will leave the European Union.

“We need to take control of our borders,” he says.

But these sentiments are, at the moment, as valuable to him as his personal views on Her Majesty’s latest hat. He is more keen to know how the rest of the British electorate is inclined so he can win some money.

Mr. Howe is standing inside a William Hill gambling shop near Westminster, home to the British Parliament. It is midday Wednesday, less than 24 hours before British voters will go to the polls to determine the political geography of Europe. Video roulette machines flicker behind him. Televisions above him beam in dog races from hither and yon.

A single screen attracts Mr. Howe’s interest: one showing the odds on the so-called Brexit. They have in recent days tilted heavily toward Britain sticking with the European Union. A bet of 9 pounds to remain (about $13.24) fetches a profit of only £2 (less than $3). By late afternoon, the odds slide even more, showing an 80 percent chance that voters will maintain Britain’s place in the European Union.