Due to its potential to support high data rates at low latency with reasonable interference isolation, millimeter-wave (mmWave) communications has emerged as a promising solution for wireless personal-area networks (WPAN) and an enabler for emerging applications such as high-resolution untethered virtual reality. At mmWave, signals are prone to blockage by objects in the environment, including human bodies. Most mmWave systems utilize directional antennas in order to overcome the significant path loss. In this paper, we consider the effects of blockage and antenna directivity on the performance of a mmWave WPAN. Similar to related work, we assume that the interferers are in arbitrary locations and the blockages are drawn from a random point process. However, unlike related work that assumes independent blocking, we carefully account for the possibility of correlated blocking, which arises when two interferers are close to each other and therefore an obstruction that blocks the first interferer may likely block the second interferer. Closed form expressions for the blockage correlation coefficient and the distribution of the SINR are provided for the case of two dominant interferers and a fixed number of blockages drawn from a binomial point process. Finally, the effects of antenna directivity and the spatial randomness of the interferers are taken into account, resulting in SINR curves that fully account for correlated blocking, which are compared against curves that neglect correlation. The results provide insight into the validity of the commonly held assumption of independent blocking and the improved accuracy that can be obtained when the blocking correlation is taken into account.