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On a black table, many items are displayed: cards fanned out, rollingdice, and poker chips.

Trying to have a baby with the aid of modern reproductive technology can feel like visiting a gambling parlor with the highest possible stakes. So consider the pitch that many fertility clinics now put in front of people like Kristi and Carleton Chambers.

After several miscarriages, the Leesburg, Va., couple took their doctors up on an offer to hand over $50,000 — $20,000 more than they might have paid for the in vitro fertilization and other services that they needed at the time. In return, the clinic promised multiple procedures until they gave birth, and if it didn’t work, they would get a full refund.

The catch? If they made a baby on the first try, the practice would keep all their money. That is exactly what happened — to their great joy. After their baby boy was born, the couple eventually signed up for a similar deal and ended up with twins.

Welcome to the fertility casino, which frequently presents the rarest of scenarios: A commercial entity offers a potentially money-losing proposition to customers in exchange for...