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People told Isabel Sanchez that the housing lottery was a waste of time. The odds were too long. Her family would never win. Happy endings are rare in San Francisco’s housing market for anyone who can’t afford luxury rents.

But she and her family were facing a no-fault eviction last fall; the house where they rented an in-law unit had been sold to new owners, who wanted their mother to move in. So Ms. Sanchez applied for the city-run lottery for Natalie Gubb Commons, a new 95-unit affordable housing development downtown open to households making up to 50 percent of the area median income.

Ms. Sanchez and her family were among 6,580 households that applied for the property, whose lottery and aftermath The New York Times chronicled through this spring. Such lotteries are a regular feature of housing policy in high-cost cities. Far more people need — and qualify for — public housing, housing vouchers or below-market-rate apartments than the assistance available. And so cities, housing authorities and developers often dole out what they have by drawing numbers.

Ms. Sanchez pulled No. 16. In April, her family moved into a new three-bedroom apartment renting for a fraction of market rate. That they won the first lottery they ever entered — while some people apply, and fail, for years — shows how utterly random the results are.