I personally do not believe we evolved from apes.

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Shhh, don't tell.

Anyway, mutations in DNA can change an organism's traits, which can affect how well the organism survives in its environment.  Organisms which are better adapted to their environment live and become more common than, or replace, their ancestor, while harmful mutations make it more difficult for the organism to live.

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What has DNA and Genetics got to do with darwins theory as evidance?

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I'll try to do this simply:

Genes are made up of DNA, which is commonly referred to as a blueprint for life. Genes determine our traits. Traits, like what skin color you have, hair, eyes, the length of your arms, etc matter in how you go about your life. Some traits are better then others, for instance the ability to process more types of food. That creates an advantage. The species that has the most advantages and can adapt will be selected by natural selection and will propagate. Over time species change with their enviroments and new species arise. By tracing genes we can determine our closest relatives and when they and us split from a common ancestor. We share the vast majority of our genes with chimps, meaning that humans and chimps came from the same ancestors relatively recently (couple million years). Mankind shares relatively fewer genes with starfish, meaning our common ancestor lived hundreds of millions of years ago.

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"I give the probability of this working at zero" - Dan DeMatteo