In-network caching is a fundamental mechanism advocated by information-centric networks (ICNs) for efficient content delivery. However, this new mechanism also brings serious privacy risks due to cache snooping attacks. One effective solution to this problem is random-cache, where the cache in a router randomly mimics a cache hit or a cache miss for each content request/probe. In this paper, we investigate the effectiveness of using multiple random-caches to protect cache privacy in a multi-path ICN. We propose models for characterizing the privacy of multi-path ICNs with random-caches, and analyze two different attack scenarios: 1) prefix-based attacks and 2) suffix-based attacks. Both homogeneous and heterogeneous caches are considered. Our analysis shows that in a multi-path ICN an adversary can potentially gain more privacy information by adopting prefix-based attacks. Furthermore, heterogeneous caches provide much better privacy protection than homogeneous ones under both attacks. The effect of different parameters on the privacy of multi-path random-caches is further investigated, and the comparison with its single-path counterpart is carried out based on numerical evaluations. The analysis and results in this paper provide insights in designing and evaluating multi-path ICNs when we take privacy into consideration.