There is a subtlety here that must be recognized. Since each ovum can contain either of two chromosomes, with a similar situation for the sperm, there is a random, statistical, quality to the observed results of the any given union of ovum and sperm. If one species of pea has a chromosome pair, AB, and another species of pea has a chromosome pair, ab, then these chromosomes segregate to form gametes with single chromosomes, A, B, a, b. If we label the possible ovum chromosomes A and B, and the sperm chromosomes are labeled a and b, then it is clear that the union of ovum and sperm could yield chromosome pairs Aa or Ab or Ba or Bb, giving 4 possible results. So the observable results of a sexual cross are random selections from 4 objects, though the genetic process or rules that govern the process are not random or statistical, but quite rigidly determined. The traits of the ovum-sperm union of chromosomes A and b are deterministic and predictable as any physical law can be. One simply does not know which of the possible pairs of chromosomes will result from the union of a randomly chosen ovum and sperm. So the genetic principle is determined but the observation will be a random selection of one of the possible pairs, and statistical in nature. In the same way, the properties of the quantum mechanical systems of physics are exactly determined, but the observations appear to be random and statistical in nature. The ``uncertainty'' that seems so fascinating, particularly to non-physicists, is a property produced by the need to make a measurement of the physical system in a finite time, and not a property of the physical process. This can be stated in another manner that may forestall some objections to this point of view. Granted, the Schrodinger wave functions are a probability distribution, the quantum state energies are parameters, numbers, and not a statistical parameter. The energies observed for a collection of atoms, say, will be statistically distributed because of the different laboratory frame velocities of the atoms, or because of the different perturbations of each atom by its neighbors, or because of the finite natural lifetimes of the quantum states. The atom state energies, like the distribution of chromosomes in the egg cell or the sperm cell, are well defined parameters. It is the measurable properties of a collection of atoms, or fertilized eggs that are, in their totality, a statistically distributed description of all possible configurations of the observed system.
It was clear to Mendel that to generate the statistical data of sufficient utility to allow him to make inferential deductions about the properties, in combination, of the genetic entities in the ovum and sperm would require a carefully selected medium with which to work, and numerous progeny from which to build the statistical data base. The medium he chose was varieties of garden peas -- Pisum sativum, quadratum, and umbellatum -- and the traits he selected to track through several hybrid generations were 7: namely, 1. Seeds smooth or deeply wrinkled. 2. Cotyledons, within the seed, tinted yellow or green. 3. Seed coat white or colored. 4. Ripe pods smooth or constricted between seeds. 5. Unripe pods color yellow or green. 6. Flower position terminal or distributed along stem. 7. Stem length long or short.
Mendel observed other differentiating properties, but did not use them. For example, a differentiating characteristic is the color of the flowers, white or colored. His observations showed him that this trait was linked to trait 3., since the white flowers occurred constantly correlated with white seed coat color. In Mendel's opinion the 7 traits he chose were independent traits that reproduced constantly within their given variety. They would be called unlinked, or autosomal dominant and recessive, traits in modern terminology. Traits determined by different genes on a single chromosome would be transmitted together, if that chromosome is the one participating in the ovum-sperm union; hence, they appear together, as the white seed and white flowers do, and are called ``linked''. It is also known now that peas have 7 pairs of chromosomes. It was at first assumed that only 7 unlinked traits could exist, one for each chromosome. In fact, a reference from a time when linkage was not well understood, points out that if ...a cross involving 8 factors did give theoretical expectation for the independent Mendelian segregation of eight pairs of factors, the chromosome theory, as at present held, would either be disproven or modified.[7] Although it is true that Mendel could not have found more than 7 traits that could have been unlinked by virtue of the fact that the genes determining those traits reside on separate chromosomes it turns out that the traits he studied were confined to only 4 chromosomes; chromosome 1 for traits 2 and 3, chromosome 4 for traits 4, 6, and 7, chromosome 5 for trait 5, and chromosome 7 for trait 1.[8] [9][10] The disappearance of linking of traits 2 and 3, say, on chromosome 1 arises from the fact that the genes are so far removed from each other that a process of recombination during pairing of chromosomes randomly redistributes the remote genes between the 2 chromosomes independently. [11] This is so because the probability of the two genes remaining on the same chromosome of a pair is inversely proportional to the distance between them. On chromosome 4 it turns out that there is linkage between traits 4 and 7. However, Mendel is not known to have studied the simultaneous segregation of these two traits. [12] That Mendel came near to the limits of the number of independent variables of his system through careful study could give one confidence in the credibility of his work. One finds, however, in the literature also the opinion that his limiting himself to the number of distinct parameters equal to what now proves to be the number of chromosomes was a stroke of luck. As we have seen there is no causal relationship between the 7 in the number of Mendel's traits and the 7 in the number of chromosomes, the comments are fatuous, and luck was not at play there. A full discussion of this process of selecting 7 independently inheritable traits in terms of the dogma of modern genetics leads one into matters of exquisite complexity that are far beyond the reach of the present author, and not needed for the story to be told here. A look at some references cited here will show how diverse the various discussions of this matter of linkage can be. [13][14][15]