Lamarck’s ideas of evolution included many different components. The idea that acquired traits could be inherited was a guess about the mechanism of how traits originate and are passed on, one which has proved to be largely incorrect for biological evolution (though fairly accurate for learned behaviors). He also thought there was continual abiogenesis of simple forms and steady progression towards more “advanced”, though with environmental influences contributing to diversity rather than everything being on the same straight line path. He also made particular suggestions about what organisms might be closer to the transitions between major groups. Today, “Lamarckian” often just refers to the inheritance of acquired features instead of the more core aspects of his model.
DNA methylation and resulting inactivation of particular genes can be inherited across a few generations within humans, though things with shorter generation times are much easier to study.
Consider the sequence of digits:
841971693997105
It is “random” in the sense that the source shows an overall equal probability of any one digit at each place in the sequence. But it can be considered “nonrandom” in that it is a sequence from pi.
Similarly, mutations are “random” with respect to whether a particular mutation will be useful. Our best models for “will mutation X happen?” involve a probability. Organisms cannot notice that having a lactase enzyme would be handy and increase their chance of having that mutation. However, certain mutations are more likely than others. Many organisms, including mammals, have high mutation rates in certain genes where having new features is useful (particular components of the immune system, for example). This enhances the chance of both useful and useless mutations, but natural selection can favor the useful ones once they happen. Similar, a report of “Lamarckian” evolution in bacteria took bacteria that had an artificially disabled gene and put it in a situation where repairing the gene (a reverse mutation from the change made by the researchers) would be useful. Some eventually had a mutation that repaired the gene. The gene was re-disabled and the bacteria put back through the same process. The second time around, the repair mutation happened faster in some of the bacteria. But it turned out that those bacteria had mutations in other genes, involved in error checking. The first step of the experiment had inadvertently selected for bacteria with a higher rate of mutation. The same process may happen naturally. If there’s a lot of competition, then sticking with the existing successful formula is likely to be a good plan; evolution will favor not changing, and higher mutation rate is a disadvantage. But if competition is not intense, then alternatives have more of a chance to get established. The lower competition could be because resources are abundant, or it could be because everyone’s struggling just to get by and not able to compete. The latter might happen with a change in the environment.
Is the mutation to repair the lab-disabled gene “random”? It is random in the sense that the probability of it happening is, as far as we can tell, best described by a certain percent chance of happening. It is no more likely than assorted useless mutations. Not all of the bacteria had that mutation. But there are certain mutations that are or aren’t possible. Mutations in an organism are starting from something that works, so the chance that the result also works is pretty good. Yet there is a chance that the new something works quite differently is also significant; small changes in sequence can make big differences in effect and conversely big sequence differences can have nearly identical effects. The utility of the mutation is tested against the environment - does the organism survive and reproduce? Thus, selection is a major non-random component. Theologically, we can assume that God is in control - also quite non-random; however, He does make use of what are, to our eyes, mathematically random processes.