Week 11 · Chapter

Chapter 11 — Mendelian Genetics

Introduction to Biology Free sample edition

The module's primary reading · ~40–50 minutes · Objective 6 (apply the principles of inheritance — segregation, independent assortment, the product rule, monohybrid, test, and dihybrid crosses, and the extensions of Mendel (incomplete dominance, codominance, multiple alleles, sex linkage, linkage, pedigrees) — to predict the outcomes of genetic crosses as probabilities), with Objective 5 (describe the eukaryotic cell cycle and the ordered phases of mitosis, compute a mitotic index, contrast mitosis with meiosis, and explain how crossing over, independent assortment, and fertilization generate genetic variation, including the 2ⁿ count) carried forward · Read it first this week; the lecture, the slides, the tutorial, the quiz, and the lab all build on it.

In plain language

Children often look a little like each parent, and sometimes a trait seems to skip a generation and come back. This week you learn the rules that explain it, rules Gregor Mendel worked out with garden peas and published in 1866. Each parent hands down one allele (one version of a gene) for each trait, and chance decides which one. With a simple grid called a Punnett square (a table that lists every way the parents' versions can combine), you can predict the odds of each kind of offspring before any are born. You will also see why those odds describe many offspring well and one small family poorly.

Why you already care about this

Picture a family photo. Two parents both have one look for some trait, and one of their children has a different look that neither parent shows. Relatives lean in and ask where it came from. The honest reply is that it was there all along, hidden in both parents, and it showed up in the child by chance.

That puzzle was cracked in a monastery garden. Gregor Mendel crossed garden peas and published Experiments in Plant Hybridization in 1866. His plants gave 5,474 round seeds and 1,850 wrinkled seeds in one generation, close to 3 round for every 1 wrinkled. A 3:1 pattern, a grid of 4 boxes, and a grid of 16 boxes carry this whole week.

You meet these rules more often than you might think. A gardener choosing which two tomato plants to cross, a dog breeder reading a litter, and a genetic counselor talking with a family all use the same odds. This chapter turns those odds into steps you can follow.

By the end of this chapter you will be able to:

tell a gene from an allele, a genotype from a phenotype, and homozygous from heterozygous;
explain why dominant does not mean common;
state the law of segregation and tie it to meiosis;
fill a Punnett square for Tt × Tt and read the genotype ratio and the phenotype ratio;
use the product rule and the sum rule to find the probability of an offspring type;
use a test cross to tell TT from Tt;
work a two-gene cross to the 9:3:3:1 ratio, by the grid and by the product rule;
compare a coin-toss run with the expected counts and explain why more trials land closer to the odds.

Carry this sentence with you: a ratio is odds over many offspring, not a promise.

Where we are

Already covered: Week 9 showed how mitosis copies a cell, and Week 10 showed how meiosis halves the chromosome number and shuffles chromosomes into gametes. Week 1 taught you to compare a result with a prediction and to say evidence supports a claim rather than proves it.
This week: the vocabulary of inheritance, Mendel's two laws, the Punnett square, the product rule and the sum rule, the test cross, the two-gene cross, and a coin-toss model of a cross.
How they connect: the laws you learn this week are what meiosis does to alleles; the grids are a way to list every gamete meiosis can make.
Where we're going next: Week 12 bends Mendel's simple rule with incomplete dominance, codominance, multiple alleles, sex linkage, and pedigrees.

Week 10 is the one this chapter leans on hardest. You learned there that a body cell carries chromosomes in homologous pairs, one chromosome of each pair from each parent, and that meiosis I pulls the two members of each pair apart into different cells. You also learned that each pair lines up at random, independently of the other pairs. Hold on to both facts. This week they become Mendel's two laws.

1. Learn the vocabulary: gene, allele, genotype, phenotype

Think of a cookbook passed down in a family. One page is the recipe for bread. Two copies of that page might differ a little: one says to add honey, the other says to leave it out. The page is the same page, but the versions differ. Genes work the same way.

A gene is a stretch of DNA that carries the instructions for a trait, such as seed shape in a pea plant.
An allele is one version of a gene; a body cell carries two alleles of each gene, one on each chromosome of a homologous pair, and the two can match or differ.

Biologists write alleles with letters. Picture a plant with a dominant allele, T (the tall allele), and a recessive allele, t (the short allele). This plant and its trait are illustrative, a model for practice rather than a real measured species.

A dominant allele is an allele whose trait shows whenever it is present, even with a different allele beside it; it is written as a capital letter, such as T. Dominant does not mean more common or stronger.
A recessive allele is an allele whose trait shows only when it is the only version present, as in tt; it is written as a lowercase letter, such as t.

Now name the pair a plant carries and the trait it shows.

A genotype is the pair of alleles an organism carries for a gene, such as TT, Tt, or tt; it is the recipe rather than the visible result.
A phenotype is the trait an organism shows, such as round or wrinkled seeds; TT and Tt give the same phenotype because the dominant allele shows.
Homozygous means carrying two matching alleles of a gene, either two dominant (TT) or two recessive (tt).
Heterozygous means carrying two different alleles of a gene, one dominant and one recessive (Tt).

Worked example — read three genotypes. You sort the three possible genotypes of the illustrative plant.

TT: two matching alleles, so homozygous. T is present, so the phenotype is tall.
Tt: two different alleles, so heterozygous. T is present, so the phenotype is tall.
tt: two matching alleles, so homozygous. No T is present, so the phenotype is short.

Two of the three genotypes look tall. You cannot tell TT from Tt by looking at the plant. That is the trap to remember: TT and Tt look the same, but they cross differently, as Section 7 will show.

Misconception check. Students often use genotype and phenotype as if they meant the same thing. The words sound alike and both describe a trait. The cure is the kitchen picture: the genotype is the recipe, and the phenotype is the cake. Two different recipes, TT and Tt, can bake the same-looking cake.

2. Cure the dominant-means-common myth

Imagine a crowd at a concert. Suppose one allele in that crowd is dominant and another is recessive. It is natural to guess that the dominant one must be in most people. It need not be.

Dominant describes what happens inside one organism with two alleles. It says that when T sits beside t, the T trait is the one you see. It says nothing about how many people or plants in a population carry T. A recessive allele can be far more common in a population than the dominant one. A dominant allele can be rare.

Worked example — two separate questions. You ask two questions about the same illustrative plant population.

Question 1, inside one plant: a Tt plant shows which trait? T is present, so it shows the dominant trait.
Question 2, across the whole population: which allele is more common? Dominance cannot tell you. You would have to count the alleles in the population.

The first question is about dominance. The second is about frequency. They have different answers because they ask different things.

Misconception check. Students often think dominant means more common or stronger. The word "dominant" sounds like winning, so it feels like the dominant allele should spread and take over. The cure is the definition: dominant means the allele shows when present, nothing more. A recessive allele can be the common one.

3. Meet Mendel and his peas

Picture a garden bed of pea plants, some with round seeds and some with wrinkled ones. Anyone can see that a trait passes down. Mendel's work comes with counts, and a count lets you see a pattern.

Gregor Mendel (1822–1884) crossed garden peas at an Augustinian monastery in Brno. He published Experiments in Plant Hybridization in 1866. Few people noticed the paper until it was rediscovered in 1900.

Geneticists name the generations of a cross. The parents are the P generation. Their offspring are the F1 generation.

The F2 generation is the second generation of offspring in a cross: the offspring of the F1 generation, which are themselves the offspring of the original parents (the P generation).

For seed shape, Mendel's F2 plants gave 5,474 round seeds and 1,850 wrinkled seeds.

Worked example — reading Mendel's count. You check how close the count is to 3 to 1.

Divide round by wrinkled: 5,474 ÷ 1,850 ≈ 2.96.
2.96 is close to 3, so there were about 3 round seeds for every 1 wrinkled seed.
In words: his F2 result landed near a 3:1 ratio, not exactly on it.

Honesty note: the raw counts are history, told here so you can see a real result sit near the ratio. The fact to remember is the 3:1 ratio, and that is what this course grades.

Misconception check. Students sometimes think a scientific ratio has to come out exact, so 5,474 to 1,850 looks like a failed result. The idea is tempting because textbook ratios are tidy numbers. The cure is to remember that a ratio describes odds, and a real count lands near the odds, not on them. A count near 3 to 1 supports the rule.

4. State the law of segregation

Think about a deck of two cards, one marked T and one marked t. If you hand someone one card, they get T or t, never both. Each parent is like that deck for each gene.

The law of segregation is Mendel's rule that each parent passes exactly one allele of each gene to each offspring, because the two alleles separate into different gametes when homologous chromosomes part in meiosis I.

This is where Week 10 pays off. A Tt parent carries T on one chromosome of a homologous pair and t on the other. In meiosis I the pair separates. Half of the gametes get the chromosome with T, and half get the chromosome with t.

Worked example — gametes from three genotypes. You list the gametes each genotype can make.

TT: both chromosomes carry T, so every gamete carries T.
Tt: one chromosome carries T and one carries t, so half the gametes carry T and half carry t. P(T) = 1⁄2 and P(t) = 1⁄2.
tt: both chromosomes carry t, so every gamete carries t.

Each gamete carries one allele of the gene, because each gamete gets one chromosome from the pair.

Misconception check. Students often say each parent passes both alleles to a child. The idea is tempting because a child does end up with two alleles. The cure is to count where they come from: one from the egg and one from the sperm. Each parent gives one, because meiosis I splits the pair.

5. Fill a Punnett square

A Punnett square is a grid that lists one parent's possible gametes across the top and the other parent's down the side, so each box shows one equally likely offspring genotype. It is a bookkeeping tool. It makes sure you count every combination once.

A monohybrid cross is a cross that follows one gene, such as Tt × Tt, which gives a genotype ratio of 1:2:1 and a phenotype ratio of 3:1.

Worked example — the Tt × Tt square. You cross two heterozygous plants, Tt × Tt, where T (the dominant allele) shows the dominant trait and t (the recessive allele) shows the recessive trait only in tt.

Gametes: by the law of segregation, each parent makes gametes with T or with t, half and half.
Set up: write T and t across the top for parent 1, and T and t down the side for parent 2.
Fill the four boxes, one allele from the top and one from the side.

Punnett square for a cross of two Tt parents: one parent's gametes across the top, the other's down the side, offspring genotypes in the boxes
	T (parent 1)	t (parent 1)
T (parent 2)	TT	Tt
t (parent 2)	tT	tt

Count the genotypes: tT is the same genotype as Tt, so the boxes hold 1 TT, 2 Tt, and 1 tt.
Genotype ratio: 1 TT : 2 Tt : 1 tt, written 1:2:1, so P(TT) = 1⁄4, P(Tt) = 2⁄4 = 1⁄2, and P(tt) = 1⁄4.
Phenotype ratio: TT and Tt both show the dominant trait, so 1 + 2 = 3 boxes are dominant and 1 box is recessive, a 3:1 ratio; P(dominant) = 3⁄4.

In words: a cross of two heterozygous parents gives three dominant-looking offspring for every one recessive-looking offspring, on average. That is Mendel's 3:1.

Misconception check. Students often give the phenotype ratio of Tt × Tt as 1:2:1. The slip is tempting because 1:2:1 is the first ratio you read off the boxes. The cure is to ask what you can see. TT and Tt look the same, so they fall into one phenotype group: 1 + 2 = 3 dominant to 1 recessive. The 1:2:1 is the genotype ratio, never the phenotype ratio.

6. Use the product rule and the sum rule

Flip one coin, and the chance of heads is 1 in 2. Flip two coins, and the chance that both land heads is smaller, 1 in 4. That everyday fact is the first of two rules that turn a Punnett square into arithmetic.

Probability is the chance that an outcome happens, written as a reduced fraction from 0 (never) to 1 (always), such as P(tt) = 1⁄4.
The product rule is the probability rule for two independent events both happening: multiply their probabilities, so a t from each of two Tt parents is 1⁄2 × 1⁄2 = 1⁄4.
The sum rule is the probability rule for either of two outcomes that cannot happen at the same time: add their probabilities, so P(TT) + P(Tt) = 1⁄4 + 1⁄2 = 3⁄4.

The product rule handles "this and that." The sum rule handles "this or that."

Worked example — the Tt × Tt cross without a grid. You find two probabilities with the two rules.

Product rule: P(tt) = P(t from parent 1) × P(t from parent 2) = 1⁄2 × 1⁄2 = 1⁄4, matching the one tt box out of four.
Sum rule: P(dominant) = P(TT) + P(Tt) = 1⁄4 + 1⁄2 = 3⁄4, matching the three dominant boxes out of four.

In words: both rules agree with the grid, and they will save you time when the grid gets big.

Odds over many, not a promise in four

A probability describes many offspring. It does not promise anything about a few. Picture a family with four children whose parents are both Tt for an illustrative trait. Many people expect exactly 3 children with the dominant trait and 1 with the recessive trait. Chance does not work that way.

Worked example — the small-family puzzle. You find the chance that all four children of two Tt parents show the dominant trait.

Each child is a separate event, with P(dominant) = 3⁄4.
Product rule for four children: 3⁄4 × 3⁄4 × 3⁄4 × 3⁄4 = 81⁄256.
81 ÷ 256 ≈ 0.32, so about 32%, roughly one such family in three.

In words: in about one family in three, all four children show the dominant trait and none shows the recessive one. The family is not breaking Mendel's rule. A family of four is too small for the odds to show.

Misconception check. Students often think 3:1 is guaranteed in a family of four, so a family of four children who are not 3 and 1 must be unusual. The idea is tempting because 3 + 1 = 4 fits so neatly. The cure is to treat each child as a new coin flip. The 3:1 ratio is reliable over many offspring; in four, many other splits are common.

Pause and reflect. Choose one of these two prompts and think it through, or jot a few sentences in your notes.

(a) Think of a family you know, real or from a story, where a trait seemed to skip a generation. How would you explain it now, using the words allele, recessive, and heterozygous?
(b) A friend says the odds of a 3:1 split must be high in every family of four. Using the small-family puzzle, how would you explain to that friend why a few children are a poor test of a ratio?

7. Run a test cross

Suppose a seed seller hands you a tall plant of the illustrative species and swears it is TT. You cannot check by looking, because TT and Tt both look tall. You can check by breeding.

A test cross is a cross between an organism that shows the dominant trait and a homozygous recessive (tt) partner, used to learn whether the unknown is TT (all offspring dominant) or Tt (offspring 1:1).

The tt partner is the key. It can only give t, so every offspring's phenotype shows which allele the unknown parent gave.

Worked example — two possible results. You fill both squares, one for each possible unknown.

If the unknown is Tt: Tt × tt gives the boxes Tt, Tt, tt, tt, so 2 Tt : 2 tt reduces to 1:1. Half the offspring are tall and half are short.
If the unknown is TT: TT × tt gives Tt in all 4 boxes. Every offspring is tall.

In words: if even one short offspring appears, the unknown parent must carry t, so it is Tt. If a large number of offspring are all tall, the unknown is very likely TT.

Caveat: an all-tall result from only a few offspring is weaker evidence than one short offspring. A Tt parent could, by chance, give T to every one of a small litter. One recessive offspring settles it; a long run of dominant ones only supports it.

Misconception check. Students often treat TT and Tt as interchangeable, because both show the dominant trait. The cure is the test cross: TT × tt gives all Tt, while Tt × tt gives 1:1. The two genotypes look alike and breed differently.

8. Add a second gene: independent assortment and the dihybrid cross

Now follow two traits at once in the illustrative plant: height, with T and t, and seed color, with Y (the yellow allele) and y (the green allele). Does the height allele a gamete gets affect the color allele it gets? For genes on different chromosome pairs, no.

The law of independent assortment is Mendel's rule that the alleles of different genes go into gametes independently of one another, because each chromosome pair lines up at random in meiosis I.
A dihybrid cross is a cross that follows two genes at once, such as TtYy × TtYy, which fills a 16-box grid and gives the phenotype ratio 9:3:3:1.

Worked example — the TtYy × TtYy cross. You work the cross two ways and check that they agree.

Gametes: each gamete gets one allele of each gene. A TtYy parent can make TY, Ty, tY, and ty, so 4 gamete types, each with probability 1⁄4.
Grid size: 4 gamete types per parent, so 4 × 4 = 16 boxes.
Count by the grid: fill all 16 boxes and sort them by phenotype. 9 boxes show both dominant traits, 3 show dominant height with the recessive color, 3 show the recessive height with the dominant color, and 1 shows both recessive traits.
Check the total: 9 + 3 + 3 + 1 = 16.
Count by the product rule: treat each gene as its own Tt × Tt cross.
Both dominant: 3⁄4 × 3⁄4 = 9⁄16.
Each mixed class: 3⁄4 × 1⁄4 = 3⁄16.
Both recessive: 1⁄4 × 1⁄4 = 1⁄16.

In words: the 16-box grid and the product rule land on the same 9:3:3:1. The product rule gets there in three lines.

Misconception check. Students often mislabel the dihybrid classes, for example calling the 9 the both-recessive group. The slip is tempting because there are four numbers to place. The cure is to build each class from its two parts. Both dominant is 3⁄4 × 3⁄4, the largest, so it is the 9. Both recessive is 1⁄4 × 1⁄4, the smallest, so it is the 1. The two 3s are the mixed classes.

9. Test the odds with coins

A coin is a fair model of a Tt parent. Heads stands for T, tails for t, and each comes up half the time, in the same way that each allele goes into half the gametes. Flip two coins at once and you have one offspring of Tt × Tt.

An expected count is the number of offspring of a kind the odds predict for a given total, found by multiplying the total by the probability, such as 100 × 1⁄4 = 25.
The law of large numbers is the pattern that results from chance events land closer to the predicted odds as the number of trials grows, so 100 coin flips sit nearer 3:1 than 4 flips do.

The coin-toss example below is illustrative, a run constructed for this course rather than a real class result.

The coin-toss example (illustrative): offspring counts from 100 two-coin flips beside the counts the odds predict
Genotype	Model run	Expected	Gap from expected
TT	23	25	−2
Tt	52	50	2
tt	25	25	0
Total	100	100	0

Worked example — reading the coin-toss run. You compare the run with the prediction, step by step.

Expected counts in 100 flips: TT = 100 × 1⁄4 = 25; Tt = 100 × 1⁄2 = 50; tt = 100 × 1⁄4 = 25; 25 + 50 + 25 = 100.
Model run: the run gave 23 TT, 52 Tt, and 25 tt; 23 + 52 + 25 = 100, so every flip was counted once.
Dominant count: 23 + 52 = 75 offspring show the dominant trait, and 25 show the recessive trait.
Observed ratio: 75:25, and 75 ÷ 25 = 3, so the run reduces to 3:1, the same as the prediction.
Gap from the odds: TT is 23 − 25 = −2 from expected, Tt is 52 − 50 = 2 from expected, and tt is 25 − 25 = 0 from expected.

In words: the genotype counts sit near 25, 50, and 25 but not on them, and the phenotype counts reduce to 3:1. A real run lands near the ratio, not on it.

This is the shape of this week's Biology Lab & Inquiry. You will flip two coins 100 times, tally the three genotypes, compare your counts with 25, 50, and 25, and explain the gaps. Your own run will differ from this one. The gradable facts are the boxes, the ratios, and the read, never one particular count.

Limitation: coins model the law of segregation well, because a fair coin gives each side half the time. They do not model everything about real crosses, which Week 12 begins to show.

Misconception check. Students often think a run that misses 25, 50, and 25 means the coins or the rule are broken. The idea is tempting because the prediction sounds exact. The cure is the law of large numbers: more trials land closer to the odds. A run of 100 that lands within a few flips of each expected count is a run that supports the rule.

Key terms

Gene — A stretch of DNA that carries the instructions for a trait, such as seed shape in a pea plant. (first used: 1. Learn the vocabulary: gene, allele, genotype, phenotype)
Allele — One version of a gene; a body cell carries two alleles of each gene, one on each chromosome of a homologous pair, and the two can match or differ. (first used: 1. Learn the vocabulary: gene, allele, genotype, phenotype)
Dominant allele — An allele whose trait shows whenever it is present, even with a different allele beside it; it is written as a capital letter, such as T. (first used: 1. Learn the vocabulary: gene, allele, genotype, phenotype)
Recessive allele — An allele whose trait shows only when it is the only version present, as in tt; it is written as a lowercase letter, such as t. (first used: 1. Learn the vocabulary: gene, allele, genotype, phenotype)
Genotype — The pair of alleles an organism carries for a gene, such as TT, Tt, or tt; it is the recipe rather than the visible result. (first used: 1. Learn the vocabulary: gene, allele, genotype, phenotype)
Phenotype — The trait an organism shows, such as round or wrinkled seeds; TT and Tt give the same phenotype because the dominant allele shows. (first used: 1. Learn the vocabulary: gene, allele, genotype, phenotype)
Homozygous — Carrying two matching alleles of a gene, either two dominant (TT) or two recessive (tt). (first used: 1. Learn the vocabulary: gene, allele, genotype, phenotype)
Heterozygous — Carrying two different alleles of a gene, one dominant and one recessive (Tt). (first used: 1. Learn the vocabulary: gene, allele, genotype, phenotype)
F2 generation — The second generation of offspring in a cross: the offspring of the F1 generation, which are themselves the offspring of the original parents. (first used: 3. Meet Mendel and his peas)
Law of segregation — Mendel's rule that each parent passes exactly one allele of each gene to each offspring, because the two alleles separate into different gametes when homologous chromosomes part in meiosis I. (first used: 4. State the law of segregation)
Punnett square — A grid that lists one parent's possible gametes across the top and the other parent's down the side, so each box shows one equally likely offspring genotype. (first used: 5. Fill a Punnett square)
Monohybrid cross — A cross that follows one gene, such as Tt × Tt, which gives a genotype ratio of 1:2:1 and a phenotype ratio of 3:1. (first used: 5. Fill a Punnett square)
Probability — The chance that an outcome happens, written as a reduced fraction from 0 (never) to 1 (always). (first used: 6. Use the product rule and the sum rule)
Product rule — The probability rule for two independent events both happening: multiply their probabilities. (first used: 6. Use the product rule and the sum rule)
Sum rule — The probability rule for either of two outcomes that cannot happen at the same time: add their probabilities. (first used: 6. Use the product rule and the sum rule)
Test cross — A cross between an organism that shows the dominant trait and a homozygous recessive (tt) partner, used to learn whether the unknown is TT (all offspring dominant) or Tt (offspring 1:1). (first used: 7. Run a test cross)
Law of independent assortment — Mendel's rule that the alleles of different genes go into gametes independently of one another, because each chromosome pair lines up at random in meiosis I. (first used: 8. Add a second gene: independent assortment and the dihybrid cross)
Dihybrid cross — A cross that follows two genes at once, such as TtYy × TtYy, which fills a 16-box grid and gives the phenotype ratio 9:3:3:1. (first used: 8. Add a second gene: independent assortment and the dihybrid cross)
Expected count — The number of offspring of a kind the odds predict for a given total, found by multiplying the total by the probability. (first used: 9. Test the odds with coins)
Law of large numbers — The pattern that results from chance events land closer to the predicted odds as the number of trials grows. (first used: 9. Test the odds with coins)

Summary

A gene is the instruction; an allele is one version of it.
Genotype is the allele pair you carry; phenotype is the trait you show.
TT and Tt look the same but cross differently.
Dominant means the allele shows when present, not that it is common.
Mendel published in 1866; the paper was rediscovered in 1900.
Segregation: each parent passes one allele, because meiosis I splits each pair.
Tt crossed with Tt gives genotypes 1:2:1 and phenotypes 3:1.
Multiply for "and" (product rule); add for "or" (sum rule).
A test cross with tt tells TT (all dominant) from Tt (1:1).
Two genes on different chromosome pairs assort independently and give 9:3:3:1.
A ratio is odds over many offspring; more trials land closer to it.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

In an illustrative rabbit, the allele B gives black fur and is dominant to b, which gives brown fur. A rabbit breeder crosses a Bb rabbit with a bb rabbit. Fill the Punnett square and give the genotype ratio, the phenotype ratio, and P(brown).
Two Bb rabbits from the same illustrative line have a litter. Use the product rule to find P(bb) and the sum rule to find P(black). Then find the expected number of brown kits in 40 kits.
A community garden plot holds an illustrative flower with a dominant purple allele, D, and a recessive white allele, d. A purple flower is crossed with a white flower, and 3 of the 8 offspring are white. What is the purple parent's genotype, and how do you know?
A friend says that because the dihybrid ratio is 9:3:3:1, a cross with 16 offspring must give exactly 9, 3, 3, and 1. Explain what is wrong with that claim, using the law of large numbers.

Answers (worked reasoning — read after attempting)

Bb makes B and b gametes; bb makes only b. The four boxes are Bb, Bb, bb, bb. Genotypes: 2 Bb : 2 bb, which reduces to 1:1. Phenotypes: Bb is black and bb is brown, so 2 black : 2 brown, which also reduces to 1:1. P(brown) = 2⁄4 = 1⁄2. This is a test cross, and a heterozygous parent gives the 1:1 result.
Product rule: P(bb) = 1⁄2 × 1⁄2 = 1⁄4, because each parent must give b and the two events are independent. Sum rule: P(black) = P(BB) + P(Bb) = 1⁄4 + 1⁄2 = 3⁄4. Expected brown kits in 40: 40 × 1⁄4 = 10. In words, about 10 of 40 kits should be brown on average, though a real litter count will land near 10, not always on it.
The purple parent is Dd. The white parent is dd, so it can give only d. A white offspring is dd, so it must have received d from the purple parent too. A DD parent could give only D, which would make every offspring purple. One white offspring is enough to show the purple parent carries d.
The 9:3:3:1 ratio is odds over many offspring, so 16 offspring will usually land near 9, 3, 3, and 1 but not exactly on them. Each offspring is a separate chance event, and in a small group chance can push the counts several places off. The law of large numbers says the counts land closer to the odds as the number of offspring grows, so 1,600 offspring would sit much nearer the ratio than 16 do.

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, and the quiz; next week, incomplete dominance and codominance bend Mendel's simple rule.

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