Exact probabilities for typical ranks of 2 × 2 × 2 and 3 × 3 × 2 tensors

We show that the probability to be of rank 2 for a 2× 2× 2 tensor with elements from a standard normal distribution is π/4, and that the probability to be of rank 3 for a 3× 3× 2 tensor is 1/2. In the proof results on the expected number of real generalized eigenvalues of random matrices are applied. For n × n × 2 tensors with n ≥ 4 we also present some new aspects of their rank.