We show that neural networks with access to randomness can outperform deterministic networks by using amplification. We call such networks Coin-Flipping Neural Networks, or CFNNs.We show that a CFNN can approximate the indicator of a d-dimensional ball to arbitrary accuracy with only 2 layers and O(1) neurons, where a 2-layer deterministic network was shown to require Omega(e^d) neurons, an exponential improvement.We prove a highly non-trivial result, that for almost any classification problem, there exists a trivially simple network that solves it given a sufficiently powerful generator for the network's weights.Combining these results we conjecture that for most classification problems, there is a CFNN which solves them with higher accuracy or fewer neurons than any deterministic network.Finally, we verify our proofs experimentally using novel CFNN architectures on CIFAR10 and CIFAR100, reaching an improvement of 9.25% from the baseline.