In the paper, the effect of spontaneous Brillouin scattering (SpBS) is analyzed as a noise source in distributed acoustic sensors (DAS). The intensity of the SpBS wave fluctuates over time, and these fluctuations increase the noise power in DAS. Based on experimental data, the probability density function (PDF) of the spectrally selected SpBS Stokes wave intensity is negative exponential, which corresponds to the known theoretical conception. Based on this statement, an estimation of the average noise power induced by the SpBS wave is given. This noise power equals the square of the average power of the SpBS Stokes wave, which in turn is approximately 18 dB lower than the Rayleigh backscattering power. The noise composition in DAS is determined for two configurations, the first for the initial backscattering spectrum and the second for the spectrum in which the SpBS Stokes and anti-Stokes waves are rejected. It is established that in the analyzed particular case, the SpBS noise power is dominant and exceeds the powers of the thermal, shot, and phase noises in DAS. Accordingly, by rejecting the SpBS waves at the photodetector input, it is possible to reduce the noise power in DAS. In our case, this rejection is carried out by an asymmetric Mach-Zehnder interferometer (MZI). The rejection of the SpBS wave is most relevant for broadband photodetectors, which are associated with the use of short probing pulses to achieve short gauge lengths in DAS.