Suppose you contain one copy of the gene G. You must have received it either from your father or from your mother (for convenience we can neglect various infrequent possibilities - that G is a new mutation, that both your parents had it, or that either of your parents had two copies of it). Suppose it was your father who gave you the gene. Then every one of his ordinary body cells contained one copy of G. Now you will remember that when a man makes a sperm he doles out half his genes to it. There is therefore a 50 per cent chance that the sperm that begot your sister received the gene G. If, on the other hand, you received G from your mother, exactly parallel reasoning shows that half of her eggs must have contained G; once again, the chances are 50 per cent that your sister contains G. This means that if you had 100 brothers and sisters, approximately 50 of them would contain any particular rare gene that you contain. It also means that if you have 100 rare genes, approximately 50 of them are in the body of any one of your brothers or sisters.
You can do the same kind of calculation for any degree of kinship you like. An important relationship is that between parent and child. If you have one copy of gene H, the chance that any particular one of your children has it is 50 per cent, because half your sex cells contain H, and any particular child was made from one of those sex cells. If you have one copy of gene J, the chance that your father also had it is 50 per cent, because you received half your genes from him, and half from your mother. For convenience we use an index of relatedness, which expresses the chance of a gene being shared between two relatives. The relatedness between two brothers is J, since half the genes possessed by one brother will be found in the other. This is an average figure: by the luck of the meiotic draw, it is possible for particular pairs of brothers to share more or fewer genes than this. The relatedness between parent and child is always exactly 1.
It is rather tedious going through the calculations from first principles every time, so here is a rough and ready rule for working out the relatedness between any two individuals A and B. You may find it useful in making your will, or in interpreting apparent resemblances in your own family. It works for all simple cases, but breaks down where incestuous mating occurs, and in certain insects, as we shall see.
First identify all the common ancestors of A and B. For instance, the common ancestors of a pair of first cousins are their shared grandfather and grandmother. Once you have found a common ancestor, it is of course logically true that all his ancestors are common to A and B as well. However, we ignore all but the most recent common ancestors. In this sense, first cousins have only two common ancestors. If B is a lineal descendant of A, for instance his great grandson, then A himself is the 'common ancestor' we are looking for.
Having located the common ancestor(s) of A and B, count the generation distance as follows. Starting at A, climb up the family tree until you hit a common ancestor, and then climb down again to B. The total number of steps up the tree and then down again is the generation distance. For instance, if A is B's uncle, the generation distance is 3. The common ancestor is A's father (say) and B's grandfather. Starting at A you have to climb up one generation in order to hit the common ancestor. Then to get down to B you have to descend two generations on the other side. Therefore the generation distance is 1 + 2 = 3.
Having found the generation distance between A and B via a particular common ancestor, calculate that part of their relatedness for which that ancestor is responsible. To do this, multiply J by itself once for each step of the generation distance. If the generation distance is 3, this means calculate 1/2 x 1/2 x 1/2 or (1/2)3. If the generation distance via a particular ancestor is equal to g steps, the portion of relatedness due to that ancestor is (1/2)g.
But this is only part of the relatedness between A and B. If they have more than one common ancestor we have to add on the equivalent figure for each ancestor. It is usually the case that the generation distance is the same for all common ancestors of a pair of individuals. Therefore, having worked out the relatedness between A and B due to any one of the ancestors, all you have to do in practice is to multiply by the number of ancestors. First cousins, for instance, have two common ancestors, and the generation distance via each one is 4. Therefore their relatedness is 2 x (1/2)4 = 1/8. If A is B's great-grandchild, the generation distance is 3 and the number of common 'ancestors' is 1 (B himself), so the relatedness is 1 x (1/2)3 = 1/8. Genetically speaking, your first cousin is equivalent to a greatgrandchild. Similarly, you are just as likely to 'take after' your uncle (relatedness = 2 x (1/2)3 = 1/4) as after your grandfather (relatedness = 1 x (1/2)2 = 1/2).
For relationships as distant as third cousin (2 x (1/2)8 = 1/128 we are getting down near the baseline probability that a particular gene possessed by A will be shared by any random individual taken from the population. A third cousin is not far from being equivalent to any old Tom, Dick, or Harry as far as an altruistic gene is concerned. A second cousin (relatedness = 1/32) is only a little bit special; a first cousin somewhat more so (1/8). Full brothers and sisters, and parents and children are very special (1/2), and identical twins (relatedness = 1) just as special as oneself. Uncles and aunts, nephews and nieces, grandparents and grandchildren, and half brothers and half sisters, are intermediate with a relatedness of 1/4.
...Estimates of relatedness are also subject to error and uncertainty. In our over-simplified calculations so far, we have talked as if survival machines know who is related to them, and how closely. In real life such certain knowledge is occasionally possible, but more usually the relatedness can only be estimated as an average number. For example, suppose that A and B could equally well be either half brothers or full brothers. Their relatedness is either 1/4 or 1/2, but since we do not know whether they are half or full brothers, the effectively usable figure is the average, 3/8. If it is certain that they have the same mother, but the odds that they have the same father are only 1 in 10, then it is 90 per cent certain that they are half brothers, and 10 per cent certain that they are full brothers, and the effective relatedness is 1/10 x 1/2 + 9/10 x 1/4 = 0.275.
Since all humanity is one species, we are all cousins of one another by definition. Every marriage is between a husband and wife that are cousins to some degree and the closeness of the relatedness in the marriage will help in determining the relatedness of everyone.