A pea-plant cross and the meaning of an expected ratio
A hypothetical pea trait is controlled by a single autosomal locus with alleles P and p. Complete dominance is explicitly assumed: PP and Pp plants have purple flowers, while pp plants have white flowers. There is no differential survival among genotypes, mating and fertilization follow the supplied Mendelian model, and offspring outcomes are independent for this exercise. Two heterozygous plants are crossed and produce 100 scored offspring: 78 purple and 22 white. A student declares the model disproved because the counts are not exactly 75 and 25. The class then considers a separate test cross between an unknown purple parent and a pp parent, and must distinguish what a white offspring would establish from what a finite run of purple offspring would show.
1. Give the expected genotype and phenotype probabilities and calculate the supplied chi-square statistic.
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The Pp by Pp cross gives PP, Pp, pp probabilities 1/4, 1/2, 1/4, so purple and white probabilities are 3/4 and 1/4. Expected counts are 75 and 25. Chi-square is (78-75) squared/75 plus (22-25) squared/25 = 0.12 + 0.36 = 0.48. Expected probabilities do not require every finite sample to match them exactly.
2. Explain the information supplied by a white test-cross offspring versus only purple offspring.
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Under the stated model, a white pp offspring requires the purple parent to supply p, so that parent must be Pp. A finite run of purple offspring is compatible with PP but does not logically prove PP: a heterozygote can by chance produce only purple offspring. For example, 20 purple outcomes from Pp by pp have probability (1/2)^20 if outcomes are independent.