The casino billboards lining America's roadways tantalize with the lure of riches. "Easy Street. It's Only a Play Away," screams one in Arizona. "$7.1 Million Every Day. We're a Payout Machine," reads another.

But how often do gamblers really win? What are the chances that a gambler will win on a single day or over a longer period? Don't bother to ask the casinos. Although they gather vast quantities of data about their customers for marketing purposes, including win and loss tallies for many regulars,...