Why does a child have the same skin colour as their parents at birth ?
Children do not always have the same skin color as their parents upon birth. Skin color is a genetic trait. A child receives half of his or her genetics from the mother and half of his or her genetics from the father. In some cases, if the parents are not of the same race, the child will have skin which is a mixture of the parent's skin color--meaning either lighter or darker than each of the parents.
For some children, it may take a while for their skin tone to develop. The pigmentation in their skin is not as developed and the child, at birth, may appear very light skinned (in regards to African American babies).
Going back to the genetic aspect of skin color, one parent may not know about their ancestry. If this is the case, a child may have received some recessive genes from both parents and the child may have a completely different skin color that the parents all together.
Actually, skin color is a polygenic and multifactorial trait. Therefore, it is very difficult to predict the color of a child's skin, based on looking at the parents. Polygenic means more than one pair of genes affects the phenotype(skin color in this case). Multifactorial means that the gene expression is influenced by factors in the environment such as--amount of sun exposure. Polygenic traits are distributed in the population as a bell-shaped curve, where most individuals skin cover will be hovering around a median color. Therefore, the extremely light complexion and extremely dark complexions will be on either end of the spectrum and the majority of people will have a complexion somewhere in between.