= Independent assortment
{title2=$\theta=\tfrac12$}
<Independent assortment> means that transmission at different <genetic loci> has <conditional independence> given the parental <genotype>. With fair <Mendelian segregation>, a parent <heterozygous> at two <genetic loci>, $AaBb$, produces $AB,Ab,aB,ab$ <gametes> with <probability> $1/4$ each. For example,
$$
P(A,B\mid AaBb)=P(A\mid AaBb)P(B\mid AaBb)=\frac12\frac12=\frac14.
$$
Here $AB$ labels an <allele> combination; it does not assert that the two <genetic loci> lie on the same <chromosome>.
The physical model is stochastic orientation of different <homologous chromosome> pairs toward the two poles of the <spindle apparatus> in <meiosis I>. One pair's orientation does not determine another's. For two pairs, one division can send $A,B$ together and $a,b$ together; another can send $A,b$ together and $a,B$ together. See <Biology 2e by Mary Ann Clark and others/The Process of Meiosis> (§11.1, Metaphase I and Figure 11.5: random orientation of chromosome pairs).
\Image[/independent-assortment-chromosome-orientation.png]
{title=Two relative <homologous chromosome> orientations generate four <allele> combinations across <meioses>}
{description=Original $AaBb$ schematic for two separate <chromosome> pairs, fair independent orientations, no <chromosomal crossover>, and symmetric divisions. Fixing $A$ at the left pole removes the duplicate obtained by reversing both poles. Each displayed orientation has <probability> $1/2$. A single division gives two copies each of two complementary combinations; all four equally likely combinations appear across divisions.}
With $k$ distinguishable <homologous chromosome> pairs, there are $2^k$ choices of parental <homologous chromosome> origin for a product at a specified pole, each with <probability> $2^{-k}$ under this model. For $k=3$, enumerate $MMM,MMP,MPM,MPP,PMM,PMP,PPM,PPP$, where M and P mark parental origins. Whole-pole reversal pairs complementary choices, leaving $2^{k-1}$ unordered partitions for $k\geq1$. For $k=2$ there are four choices for a specified pole but only the two relative orientations drawn. This counts possible products across <meioses>, not eight different products of one division when $k=3$. Identical <alleles> at scored <genetic loci> can also make different parental-origin choices observationally indistinguishable.
<Independent assortment> of different <chromosome> pairs does not imply independence of every pair of <genes>. Nearby <genetic loci> on the same <chromosome> can show <genetic linkage>. <Chromosomal crossover> can create new <haplotypes>, but does not automatically erase <genetic linkage>. See <Biology 2e by Mary Ann Clark and others/Laws of Inheritance> (§12.3, Law of Independent Assortment and Linked Genes Violate the Law of Independent Assortment). In the ordinary fair two-locus model with parental <haplotypes> $AB/ab$ and <recombination fraction> $r$,
$$
P(AB)=P(ab)=\frac{1-r}{2},\qquad P(Ab)=P(aB)=\frac r2.
$$
Thus $r=0$ gives only the two parental <haplotypes>, while $r=1/2$ gives four equal <probabilities>. A <recombination fraction> of $1/2$ alone does not establish that the <genetic loci> lie on different <chromosomes>.
A motivating experiment appears in <Experiments in Plant Hybridization by Gregor Mendel> (§9, “The Reproductive Cells of the Hybrids”). In <peas>, hybrids for round versus wrinkled <seeds> and yellow versus green <seeds> were crossed reciprocally with constant wrinkled-green <plants>. The readout was visible <seed> shape and color: hybrid seed parents gave 31 round-yellow, 26 round-green, 27 wrinkled-yellow and 26 wrinkled-green; the reciprocal cross gave 24, 25, 22 and 27.
The apparatus was controlled <pollination>: opening immature <flowers>, removing \x[stamens]{magic} with forceps, and applying selected <pollen> to the <stigma (botany)>. Two-year trials established constant parental characters; greenhouse pots checked interference from insects in outdoor beds. Reciprocal crosses tested both transmission directions. These procedures are described in <Experiments in Plant Hybridization by Gregor Mendel> (§§2–3).
With <complete dominance> at both <genetic loci>, this <test cross> makes each <phenotype> report one hybrid <gamete> combination. The ideal <expected values> are $110/4=27.5$ and $98/4=24.5$ per class. The observations support four transmitted combinations, approximately equally frequent. They do not directly image <chromosome> orientation or prove independence for all <genes>; failed development and <seed> germination also limit the inference, as §9 notes.
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