A child inherits half of the mother's DNA and half of the father's DNA. This is because in somatic cells chromosomes come in pairs, but when sperm and eggs are

A child inherits half of the mother's DNA and half of the father's DNA. This is because in somatic cells chromosomes come in pairs, but when sperm and eggs are produced, the pairs divide and each sperm or egg contains only one chromosome from each of the 23 pairs. Therefore, given that you have received one of each pair, the probability is 1/2 that your sibling received the same one of the pair. Thus, siblings share 1/2 of their DNA on average. When you get to first cousins, each line of descent from the common grandparents is diluted by 1/2 from the grandparents that they do not have in common. Thus Nth cousins share on average 1 / 2 ^(2N+1) of their DNA in the sense that the shared DNA is inherited from their common ancestor. This is the only part of their genome that would cause their phenotypes to be similar. Any other similarities between cousins is by chance or because their relatives may have married into similar ethnic groups with shared deeper ancestry.