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We perform a large-scale study to quantify just how severe the privacy leakage problem is in Facebook. As a case study, we focus on estimating birth year, which is a fundamental human attribute and, for many people, a private one. Specifically, we attempt to estimate the birth year of over 1 million Facebook users in New York City. We examine the accuracy of estimation procedures for several classes of users: (i) highly private users, who do not make their friend lists public; (ii) users who hide their birth years but make their friend lists public. To estimate Facebook users' ages, we exploit the underlying social network structure to design an iterative algorithm, which derives age estimates based on friends' ages, friends of friends' ages, and so on. We find that for most users, including highly private users who hide their friend lists, it is possible to estimate ages with an error of only a few years. We also make a specific suggestion to Facebook which, if implemented, would greatly reduce privacy leakages in its service.