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In ordinary diploid meiosis, the two copies at a genetic locus separate: each gamete receives one copy. With fair transmission, a heterozygote $Aa$ produces gametes carrying $A$ or $a$ with probability $1/2$ each. This is a transmission mechanism conditional on a parental genotype, rather than a statement about how common either allele is in a population.

For the one-locus cross $Aa\times Aa$, assume independent parental transmissions, no new mutation, and no differences in fertilization or survival between the resulting genotypes. Multiplying the two gamete probabilities gives four equally likely ordered outcomes: $AA,Aa,aA,aa$, each with probability $1/4$. Combining the two orders of the heterozygote givesOnly under the additional ideal model of complete dominance of $A$, complete penetrance, and no environmental changes to the scored characteristic do $AA$ and $Aa$ share the dominant phenotype. Its probability is then $0.25+0.50=0.75$. The inheritance calculation alone does not determine a phenotype or predict a generic disease risk.

These are probabilities, not a requirement that every four offspring contain one $AA$, two $Aa$ and one $aa$. For 100 independent offspring under the ideal model, the expected values of the three counts are $25,50,25$; the $aa$ count has a binomial distribution with standard deviation $\sqrt{100(0.25)(0.75)}\approx 4.33$. Each new transmission keeps the same conditional probabilities given the parental genotypes.

In his paper published in 1866, Mendel reported 5,474 round and 1,850 wrinkled pea seeds: $5474/7324\approx 0.7474$ round, close to the ideal $0.75$. Following descendants also distinguished two inherited classes within the round phenotype, illustrating why genotype cannot always be read from appearance.^[ref] At several genetic loci, transmission patterns additionally depend on genetic linkage and genetic recombination.

A model organism is an organism used to investigate biological processes through accessible, repeatable experiments. Its value combines experimental tractability with a carefully delimited relationship to other organisms. For example, a gene deletion in Saccharomyces cerevisiae can test whether a gene is required for growth under specified conditions; adding another gene can test functional complementation. Such interventions support causal inference more directly than merely noticing that a gene and a phenotype occur together. ^[ref]

The striking opportunity is to ask two experimentally separate questions: what does this component do here, and which part of that function survives a change of species or conditions? The yeast cross-species complementation experiment makes both questions concrete. Model organism evidence transfer requires specifying the function being compared; a useful model organism need not reproduce every feature of another organism.

A morphological species concept distinguishes species using morphology: measurable differences in form or structure. It can be applied to preserved specimens and fossils when mating cannot be observed. Similar appearance need not imply the same species, and different appearance need not imply different species. ^[ref]

An original operational example is to compare a chosen structure in specimens of comparable age, sex and rearing conditions. Suppose two apparent groups differ in size, but one group contains only juveniles. The observed difference does not yet diagnose two species. Specify the measured phenotype, variation within each sample, and whether the distinction persists across comparable specimens. A single striking specimen is a poor substitute for that comparison.

Morphology concerns biological form and structure; a structural difference is one aspect of phenotype, not automatically a species boundary.

Natural selection involves systematic differences in expected reproductive success associated with variation in phenotype. For a continuing evolutionary response, the relevant variation must be inherited: offspring must tend to preserve the differences being favored. Actual offspring counts also fluctuate by chance, producing genetic drift. Survival matters insofar as it affects contribution to future generations. Genotype fitness is therefore contextual, depending on environment and the life cycle being counted, rather than simply measuring strength or longevity. See Clark, Douglas and Choi, Biology 2e: population evolution.

In chapter IV of his 1859 book, Darwin explained natural selection through preservation of favorable variations and greater chances of survival and reproduction. See Darwin, On the Origin of Species, first edition, chapter IV. The beautiful consequence is cumulative change without foresight: inherited differences are filtered repeatedly by present conditions. Natural selection neither generates a needed mutation on demand nor guarantees progress toward a predetermined goal.

Nonparametric linkage analysis tests excess identity by descent among relatives with a shared phenotype, without specifying its complete penetrance model or disease-allele frequency. Its null still uses Mendelian segregation and the specified pedigree.

// Target: biology.bigb

For latent genotypes $G_i$ and observed phenotypes $X_i$, assume conditional phenotype independence, independent founder genotypes, and conditionally independent Mendelian transmissions. The likelihood function isThis follows by the chain rule applied to the acyclic ancestry graph, then the law of total probability over all latent genotypes. Penetrance supplies the phenotype factors and genetic recombination supplies multilocus transmission factors.

Penetrance is the probability of an expressed phenotype conditional on a specified genotype. It can depend on age, sex and environment. A fully penetrant recessive model assigns disease probability one to the recessive homozygote and zero to other genotypes; incomplete penetrance or phenocopies replace these indicators by intermediate probabilities.