Power control in mobile communications is an adaptive control system which functions routinely without direct human intervention. In this paper we examine the cost in terms of power usage of the one-parameter system identification problem of estimating the signal-to-noise ratio and thereby setting subsequent transmissions at the appropriate power level to accommodate the error detection and recovery. This is performed with an explicit formulation of the density function of the signal-to-noise ratio estimate and its dependence on the number of training samples used in this phase - it is a non-central χ² density if the noise power is known, else it is F-distributed. The objective is to develop a total cost function based on transmission power under which the price paid for adaptation becomes apparent. Our example is based on GSM mobile telephony systems and is part of a broader study into the costs associated with adaptation.