Hardy-Weinberg Practice Problems- Genetic Equilibrium Guide

What the Hardy-Weinberg Equilibrium Actually Is

Most textbooks make this sound complicated. It isn't. The Hardy-Weinberg equilibrium is simply a mathematical model that describes what happens to gene frequencies in a population when nothing is changing them. No mutations, no selection, no migration, random mating, infinite population size.

That last part is the kicker. Real populations never meet those conditions. But the model still works because it gives you a baseline. Something to compare against.

If you're solving genetics problems, you need the equation memorized and you need to know how to plug numbers in. That's what this guide does. No philosophy, no history lesson—just the math and how to use it.

The Hardy-Weinberg Equation

Here it is:

p² + 2pq + q² = 1

And the companion equation:

p + q = 1

Where:

• p = frequency of the dominant allele
• q = frequency of the recessive allele
• p² = frequency of homozygous dominant individuals
• 2pq = frequency of heterozygous individuals
• q² = frequency of homozygous recessive individuals

That's it. Everything else is just figuring out which numbers go where.

Solving Hardy-Weinberg Problems: The Method

Most problems give you one piece of information and ask for something else. You have three scenarios:

Scenario 1: You're given q² and need p, q, p², and 2pq

Example: Cystic fibrosis occurs in 1 out of 2,500 births. Find the carrier frequency.

Step 1: Calculate q²

q² = 1/2500 = 0.0004

Step 2: Find q

q = √0.0004 = 0.02

Step 3: Find p

p = 1 - 0.02 = 0.98

Step 4: Find carrier frequency (2pq)

2pq = 2 × 0.98 × 0.02 = 0.0392 ≈ 3.92%

So roughly 1 in 25 people is a carrier.

Scenario 2: You're given allele frequencies and need genotype frequencies

Example: In a population, p = 0.7 and q = 0.3. What are the expected genotype frequencies?

Step 1: Calculate p²

p² = (0.7)² = 0.49

Step 2: Calculate q²

q² = (0.3)² = 0.09

Step 3: Calculate 2pq

2pq = 2 × 0.7 × 0.3 = 0.42

Step 4: Verify

0.49 + 0.42 + 0.09 = 1.00 ✓

Scenario 3: You're given phenotype counts and need allele frequencies

Example: In a population of 1,000 individuals, 160 show the recessive trait. Find p and q.

Step 1: Count homozygous recessives

q² = 160/1000 = 0.16

Step 2: Find q

q = √0.16 = 0.4

Step 3: Find p

p = 1 - 0.4 = 0.6

Step 4: Find heterozygotes

2pq = 2 × 0.6 × 0.4 = 0.48 → 480 individuals

Hardy-Weinberg Practice Problems

Problem 1

In a population of 500 moths, 20 are light-colored (recessive phenotype). The dark allele (D) is dominant over the light allele (d). Calculate:

• q² frequency
• q allele frequency
• p allele frequency
• Number of heterozygotes

Answer:

q² = 20/500 = 0.04

q = √0.04 = 0.2

p = 0.8

2pq = 2 × 0.8 × 0.2 = 0.32

Number of heterozygotes = 0.32 × 500 = 160 moths

Problem 2

Sickle cell anemia affects 1 in 400 African Americans. Assuming Hardy-Weinberg equilibrium:

• What is the frequency of the sickle cell allele?
• What percentage of the population are carriers?

Answer:

q² = 1/400 = 0.0025

q = √0.0025 = 0.05

Carrier frequency (2pq) ≈ 2 × 1 × 0.05 = 0.10 or 10%

Problem 3

A population has 300 AA individuals, 500 Aa individuals, and 200 aa individuals. Total = 1000.

• Calculate p and q
• Is this population in equilibrium?

Answer:

Allele counts: A = (2×300) + 500 = 1100

Allele counts: a = (2×200) + 500 = 900

Total alleles = 2000

p = 1100/2000 = 0.55

q = 900/2000 = 0.45

Expected aa = q² × 1000 = (0.45)² × 1000 = 202.5

Actual aa = 200

Close enough to call it approximately in equilibrium.

Where Students Screw Up

Taking the square root wrong. Use a calculator. Don't estimate when precision matters.

Confusing phenotype and genotype frequencies. The recessive phenotype = q² only when the population is actually at equilibrium. If you don't know that, you can't assume it.

Forgetting to square p when calculating p². p² means p times p. Students sometimes just write p. That's wrong.

Using percentages instead of decimals. If you see 16%, that's 0.16 in the equation. Don't mix them.

Assuming equilibrium when the problem doesn't say so. The equation only describes equilibrium. If a population is evolving, the numbers won't fit.

Tools for Solving Hardy-Weinberg Problems

You can do this by hand. You should be able to. But if you're checking work or running through many problems, these help:

Use calculators to verify, not to replace understanding. You'll still need to set up the problems on exams.

Getting Started: Your First 5 Problems

Here's a drill to get comfortable:

1. Find q² when q = 0.3
2. Find q when q² = 0.09
3. Find 2pq when p = 0.7, q = 0.3
4. Given 25/1000 individuals show recessive phenotype, find carrier frequency
5. Given p = 0.6, calculate expected AA, Aa, and aa frequencies in a population of 500

Do these without a calculator first. Then check with one. The repetition builds the pattern recognition you need for harder problems.

When Hardy-Weinberg Doesn't Apply

The model breaks down when evolutionary forces are at work. If you calculate expected frequencies and they don't match observed frequencies, something is happening:

• Selection — some genotypes reproduce more than others
• Mutation — new alleles entering the gene pool
• Migration — individuals moving in or out
• Genetic drift — small populations, random allele frequency changes
• Non-random mating — inbreeding, assortative mating

That mismatch between expected and observed is actually useful. It tells you evolution is occurring. The equation doesn't just describe equilibrium—it helps you detect when equilibrium is broken.

What You Actually Need to Memorize

Two equations:

p + q = 1

p² + 2pq + q² = 1

And the definitions:

• p = dominant allele frequency
• q = recessive allele frequency
• p² = homozygous dominant
• 2pq = heterozygous
• q² = homozygous recessive

Everything else is plugging numbers in. You don't need to understand why the math works for solving textbook problems. You just need to recognize which value you're given and which one you're solving for.