Mendelian Genetics

The single most useful distinction in genetics is between what a cell carries and what it shows. The genotype is the pair of alleles an organism has for a gene — for example Bb. The phenotype is the observable trait that results — for example brown eyes. These are not the same thing: two organisms with different genotypes (BB and Bb) can share one phenotype, so seeing the trait does not tell you the genotype outright.

That gap is bridged by dominance. An allele is dominant when it masks its partner in a heterozygote: in Bb, the dominant B is expressed and the recessive b is hidden. “Dominant” is a statement about which allele shows up in a heterozygote — nothing more. It does not mean the allele is stronger, better, or more common in the population; a dominant allele can be rare and a recessive one can be widespread. And both alleles — dominant and recessive — are alternate versions of the same gene at the same locus, not two different genes.

Hold onto two contrasts and the rest of the topic follows: genotype vs phenotype (the alleles you carry vs the trait you show), and what dominant actually means (masks a partner in a heterozygote, not stronger or more common). Get those straight and reading a cross becomes bookkeeping.

A Punnett square is really just probability laid out in a grid. Here is the reasoning behind it — the same logic that lets you skip the grid and multiply when the crosses get big.

1. Write the genotypes, list the gametes. Start from what each parent carries. A heterozygote Bb makes two kinds of gamete, B and b, each with probability 1/2, because the two alleles separate into different gametes. Getting the gametes right is the whole setup.
2. Each fertilization is independent. Which sperm allele meets which egg allele is a fresh random event every time, with no memory of past offspring. A cross that expects 3/4 dominant offspring does not owe you a recessive one after a run of dominant ones — the next fertilization is still 3/4 to 1/4.
3. Product rule — events that happen together. To get the chance that two independent things both occur, multiply. The chance an offspring is bb is P(b from mom) × P(b from dad) = 1/2 × 1/2 = 1/4. For two genes at once, multiply each gene's probability: that is where 9:3:3:1 comes from.
4. Sum rule — either-or outcomes. To get the chance of an outcome that can happen in more than one mutually exclusive way, add. The dominant phenotype from Bb × Bb is BB or Bb: 1/4 + 2/4 = 3/4. Dominant to recessive is therefore 3:1 — the monohybrid ratio.
5. Read the ratio as expected, not guaranteed. 3:1 and 9:3:3:1 are the expected proportions over many offspring. A single litter of four will rarely be exactly 3 and 1; the ratio is a long-run average, and small samples wander around it.

The through-line: gametes carry alleles with fixed probabilities, each fertilization is independent, and you combine those probabilities by multiplying (both) or adding (either-or). The 3:1 and 9:3:3:1 ratios are just those rules worked out.

Quick reference card. For each term, read what it is and how it keeps the genotype-vs-phenotype line clear — the alleles you carry versus the trait you show.

A predicted ratio like 3:1 is a hypothesis. Chi-square is how you decide whether the counts you actually got are close enough to that prediction, or far enough off to doubt it. The statistic measures total mismatch between observed and expected:

Set up the null hypothesis first. The null hypothesis is that the offspring really do follow the expected ratio and any difference is just chance sampling. You compute the expected counts E from that ratio, compare to the observed counts O, and get a single χ² value: small when observed and expected are close, large when they diverge.

Then read p through the critical value. Compare χ² to the critical value for your degrees of freedom (categories minus one) at the 0.05 cutoff. If χ² exceeds the critical value, then p is below 0.05. Here is the part students invert: a small p (below 0.05) means REJECT the null — the deviation from the expected ratio is significant, and your predicted ratio is in trouble. A small p is bad news for the hypothesis, not a confirmation of it.

A large p is the “fits the ratio” case. When p is at or above 0.05, χ² is small, so you fail to reject the null: the data are consistent with the expected ratio. Note the careful wording — failing to reject is not the same as proving the hypothesis true; it only means you have no significant reason to abandon it. Chi-square never “confirms” a hypothesis; it can only fail to reject it, or reject it.

Keep the direction straight. Small p → big deviation → reject the null (significant). Large p → small deviation → fail to reject (consistent). Reading a small p as support for your prediction is exactly backwards.