Despite those daunting statistics, chances of winning are better than it may seem. That is because many of those who apply are disqualified, either because their earnings exceed or are below the requirements for a specific development, or because they fail to provide the necessary paperwork, including work history and tax records.

And depending on the housing project, preference may be given to those who meet certain criteria, like municipal workers, the homeless or residents of the neighborhood where the development is being built. (The latter, known as community preference, has come under attack for perpetuating racial segregation in some neighborhoods, and in the wake of a growing homeless population, some have also criticized the lottery program for not giving enough preference to the homeless.)

Those who have won housing lotteries are a diverse group, from a variety of socioeconomic, racial and geographic backgrounds. Despite their differences, however, they tend to have something in common: Many have faced challenging life events, and nearly all insist that the key to winning the lottery is determination.

“You have to know how to hustle and be on it,” said Erika Lindsey, an urban planner who spent nearly a decade applying to lotteries before winning a one-bedroom in Brooklyn, near Barclays Center. “My main advice is to be persistent.”

LIZZIE VILLAS BOAS

Mott Haven, Bronx | $251 for a one-bedroom

Lizzie Villas Boas suffers from hip dysplasia and has had 25 surgeries and four hip replacements. “I’ve had to learn to respect my pain,” said Ms. Villas Boas, 57, who is from Brazil. “I cannot change my life, and so I always try and find the happiness.”

Her first purchase for her new home reflects that. In October, when she moved into her new one-bedroom in the South Bronx, Ms. Villas Boas went to Target to fill her empty refrigerator. As she wandered the aisles, her eyes alighted on a wooden sign that read, “Give Thanks Always.”

Ms. Villas Boas, who gets around on a scooter, depends on Supplemental Security Income, and she couldn’t afford what amounted to a luxury, but she splurged on it anyway.