What is the correct sequence of events during evolution?

1. Random Variation in Reproduction

When we reproduce, we produce inexact copies of ourselves. No child will ever be 100% identical to its parent (barring cloning). These changes are usually pretty small—think hair/eye color, height, etc.—and they do not happen “for” any reason beyond the inexact nature of an organism copying itself. In other words, this variation is random. I didn’t get brown hair because my parents needed brown hair to survive, for example. 

2....

1. Random Variation in Reproduction

When we reproduce, we produce inexact copies of ourselves. No child will ever be 100% identical to its parent (barring cloning). These changes are usually pretty small—think hair/eye color, height, etc.—and they do not happen “for” any reason beyond the inexact nature of an organism copying itself. In other words, this variation is random. I didn’t get brown hair because my parents needed brown hair to survive, for example. 

2. Non-Random Natural Selection

For the most part, organisms have more children than their environment can support. Not all of them will survive long enough to reproduce—only those who are best suited to their environment. The features they inherit from their parents, as well as any random variations and mutations, will mean that some offspring will have a better chance at survival than the others. This mechanism is called natural selection, whereby random variations within a species are preserved in a non-random way. It might be random that I’m born with an extra arm, but if that helps me be a more competitive participant in the environment, then my survival due to that feature is non-random.

3. Overall Population Change and Evolution

Because I’ve survived with my extra mutant arm, there’s a stronger likelihood that my children will inherit my extra arm. Assuming that the extra arm remains a valuable tool to survive within my environment, then the number of offspring born with that feature will increase within a population. This overall change within a population is what we mean when we say evolution, which happens on the population level and is responsible for the development of new species.