How are a Bitaxe’s solo-mining block odds calculated?

Solo mining is a lottery, and the math is simple and honest. The Bitcoin network finds about 144 blocks per day. Your share of those attempts is just your hashrate divided by the whole network's hashrate:

daily block chance ≈ (your TH/s ÷ network TH/s) × 144

Worked example with round numbers: a Gamma at ~1.2 TH/s against a network of ~900 EH/s (900,000,000 TH/s) is a fraction of about 1.3 × 10⁻⁹ of the network. Times 144 blocks gives a daily probability around 1.9 × 10⁻⁷ — which works out to one expected block roughly every ~15,000 years on average.

Two things are both true: the expected time is astronomically long, and any block could still hit tomorrow — each block is an independent draw, and a small number of Bitaxe-class devices really have found blocks. That is the appeal of the lottery, stated without hype. Pooling (e.g. solo pools) does not change your expected reward; it only changes the variance and how a win is reported.