Machine learning is a promising technique for angle-of-arrival (AOA) estimation of waves impinging a sensor array. However, the majority of the methods proposed so far only consider a known, fixed number of impinging waves, i.e., a fixed source number. This paper proposes a machine-learning-based estimator designed for the case when the source number is variable and hence unknown a priori. The proposed estimator comprises a framework of single-label classifiers. Each classifier predicts if waves are present within certain randomly selected segments of the array's field of view (FOV), resulting from discretising the FOV with a certain (FOV) resolution. The classifiers' predictions are combined into a probabilistic angle spectrum, whereupon the source number and the AOAs are estimated jointly by applying a probability threshold whose optimal level is learned from data. The estimator's performance is assessed using a new performance metric: the joint AOA estimation success rate. Numerical simulations show that for low SNR (-10 dB), a low FOV resolution (2$^\circ$) yields a higher success rate than a high resolution (1$^\circ$), whereas the opposite applies for mid (0 dB) and high (10 dB) SNRs. In nearly all simulations, except one at low SNR and a high FOV resolution, the proposed estimator outperforms the MUSIC algorithm if the maximum allowed AOA estimation error is approximately equal to (or larger than) the FOV resolution.