You
Positive expected value means I should take the bet every time?
ChatGPT
Only if you can survive the variance. Repeated multiplicative bets are governed by the geometric mean, not the arithmetic one:
\[g = \prod_i (1 + r_i)^{p_i}\]A bet with positive arithmetic expectation can still have \(g < 1\), which means certain ruin over enough repetitions. That is the whole content of the Kelly criterion.
You
How long should a random id be to avoid collisions?
ChatGPT
Use the birthday approximation: collisions become likely around \(\sqrt{N}\) draws from a space of size \(N\). For a 64-bit id that is about 4 billion items, for 128 bits it is beyond anything you will generate. The practical answer for most systems is 128 random bits and no coordination.