We present a new electronic device – the single-electron bipolar avalanche transistor (SEBAT) – which allows for the detection of single charges with a bandwidth typically above 1 GHz, exceeding by far the bandwidth of other room-temperature single-electron detectors. To the best of our knowledge, it is also the first single-electron detector to be realized in a standard CMOS technology. The device is a bipolar transistor optimized for operation in the Geiger mode. Single electrons injected through the base-emitter junction trigger the avalanche breakdown of the collector-base junction, which is rapidly stopped by a quenching circuit connected to the collector. This cycle produces a quasi-digital voltage pulse which corresponds to the detection of a single electron. The intrinsic randomness of single-electron injection associated with its particular output signal make this transistor a truly probabilistic device, in that the input parameters do not deterministically control the output, but only the likelyhood of a digital pulse. A similarly pulsed operation is also characteristic of the behavior of some neurons, with which the SEBAT shares important properties. Because of the intrinsic randomness of the pulse generation process, the SEBAT allows reconstructing the power spectrum of an input function independent of the pulse rate. Therefore, it provides access to information over its full bandwidth at extremely low power consumptions, on the order of a few nW. The randomness of the single charge injection can also be used as an entropy source for a quantum random number generator. Moreover, the non-linear input characteristics of the SEBAT enable signal mixing or pulse coincidence detection.