We investigate a novel integrated sensing and communication (ISAC) system supported by pinching antennas (PAs), which can be dynamically activated along a dielectric waveguide to collect spatially diverse observations. This capability allows different PAs to view the same target from different angles across time, thereby introducing target diversity, which is a key advantage over conventional fixed antenna arrays. To quantify the sensing reliability, we adopt the outage probability as a performance metric, capturing the likelihood that the accumulated radar echo signal power falls below a detection threshold. In contrast to traditional ISAC models that assume deterministic sensing channels, we explicitly account for the look-angle dependence of radar cross-section (RCS) by modeling it as a random variable. We ensure the long-term quality-of-service (QoS) for communication users by enforcing an accumulated data rate constraint over time. We derive an exact closed-form expression for the sensing outage probability based on the distribution of weighted sums of exponentially distributed random variables. Since the resulting expression is highly non-convex and intractable for optimization, we use a tractable upper bound based on the Chernoff inequality and formulate a PA activation optimization problem. A successive convex approximation (SCA) framework is proposed to efficiently solve the formulated problem. Numerical results show that dynamically activating different PAs across time slots significantly enhances sensing reliability compared to repeatedly activating the same PA at a fixed position and conventional antenna selection schemes, respectively. These findings highlight the benefits of integrating outage-based reliability metrics and target diversity into ISAC systems using PAs.