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In this paper we study the problem of determining the optimal spatial node density for deployment of a Wireless Sensor Network (WSN) for detection of a randomly located target in a sensing field. We formulate an optimization problem for the single-hop scenario and account for factors such as the Medium Access Control (MAC) protocol that is used, the wireless channel's propagation characteristics, a randomized sleep/wake-up scheduling protocol, network coverage constraints, the energy consumed, the time to reach a decision, and the total number of nodes available. We show that the optimal node density that minimizes the average Decision Error Probability (DEP) at the CH is a function of the network parameters such as average wake-up rate, propagation path-loss exponent and the lifetime of the network. Simulations are used to study the many optimal trade-offs that are possible.