Multi-core fiber (MCF) is one of the promising space-division multiplexing technologies to increase the capacity of optical networks. MCF-based networks have two challenges. One is the inter-core crosstalk (XT) that degrades the quality of optical signals in two neighboring fiber cores. The other is network protection against link failures that cause massive data loss. One way to protect against multiple link failures is to prepare physically separated links as a backup network. Probabilistic protection improves the efficiency of protection by allowing a certain probability of protection failure. Existing studies on backup network design with probabilistic protection do not target MCF-based networks, which raises problems such as protection failure due to the inter-core XT and excessive consumption of optical resources. To address these problems, this paper proposes a XT-aware backup network design model for the MCF optical path networks. The proposed model protects the network against probabilistic multiple link failures. We adopt probabilistic protection that allows a certain probability of protection failure due to the inter-core XT and minimizes the required number of links in the backup network. We present an algorithm to satisfy the probabilistic protection requirement and formulate the model as an integer linear programming problem. We develop a heuristic approach to apply the proposed model to larger networks. Numerical results observe that the proposed model requires fewer links than the dedicated allocation model, which provisions the backup paths in the same manner as the primary paths.