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We aim to mathematically obtain the probability of recovery failure in distributed sensing and sampling for a Multiple Sensor System (MSS). In this system, sensors take samples while compressing the signal with linear projection operations using the idea of compressive sensing (CS) . In particular, we show that a bound for per-sensor measurements (PSM), the number of compressed measurements required at each sensor for good signal recovery. Our focus is to see how PSM behaves as the number of sensors increases based on the failure probability. Using the idea of Joint Typicality , we show that PSM converges to the sparsity as the number of sensors increases.