In psychology papers that do prospective power analyses (example), one often notices the convention of assuming β values (false negative rates) of 0.20, and power levels of 0.80. In other words (if I understand correctly), we are happy with a mere 80% chance of detecting true effects, and are happy to accept a whopping 20% chance of failing to detect such true effects.
Both values seem unacceptably low/high (respectively), which I was first tempted to cynically associate with claims about psychology not being a real science, namely on the low cost placed on finding false effects or on NOT finding true effects.
On second thought, however, it does stand to reason that the more you require of one criterion (e.g. low β), the more you necessarily have to compromise on the other (e.g. make do with a small power). In other words, that there be a trade-off between them.
I have trouble understanding however whether it is a coincidence that these two measures - in this case - add up to 1, and whether this has to be the case that β+power=1.