We consider content caching between a service provider and multiple cache-enabled users, using the recently proposed modified coded caching scheme (MCCS) that provides an improved delivery strategy for random user requests. We develop the optimal cache placement solution for the MCCS with arbitrary cache size by formulating the cache placement as an optimization problem to minimize the average rate during the delivery phase under random user requests. Through reformulation, we show that the problem is a linear programming problem. By exploring the properties in the caching constraints, we obtain the optimal cache placement solution in closed-form. We verify that the existing cache placement scheme obtained at specific cache sizes is a special case of our solution. Numerical studies show how the caching gain changes as the user population increases, as a result of different cache placement patterns.