There are several places during meiosis where chance plays a role. These include:
a) the positions of exchanges (crossovers);
b) the strands (chromatids) involved in the exchanges;
c) the pole choice for a centromere at the first meiotic division;
d) the pole choice for a centromere at the second meiotic division;
e) (possible) sampling of meiotic products.
Our later discussion makes extensive use of models for a) and b), so let us briefly mention the evidence for randomness at c) and d), i.e. that a given centromere has the same chance of going to either pole.
Whitehouse cites an author Carothers (1913) who examined 300 cells showing metaphase I or anaphase I of meiosis from males of a species of grasshopper heterozygous for an inequality of size of one chromosome. The single X chromosome would go to one pole, and in 146 instances, the smaller of the unequal pair went to the same pole as the X, and in 154 the larger co-segregated with the X. Whitehouse points out that there are exceptions to this random orientation of different chromosome pairs on the meiotic first-division spindle. Other evidence supporting this hypothesis comes from ordered tetrads or octads.