Introduction
A genetic description begins by specifying where in the genome a difference is being considered and which forms occur there. These distinctions matter statistically: an allele is a version of a genetic sequence at a location, while a genotype records the allele combination carried by an individual at that location. Probabilities and frequencies describe related but different things: the chance of an event for a randomly selected individual, and the proportion of individuals or allele copies observed in a population.
This tutorial sets out the basic terms and notation for one autosomal locus in a diploid organism. It does not assume any particular model for how genotypes arise or how they are distributed. In particular, allele frequencies alone do not determine genotype frequencies without additional assumptions.
Loci, Alleles, and Diploid Genotypes
For a locus with two alleles, label the alleles \(A\) and \(a\). The possible diploid genotypes are \(AA\), \(Aa\), and \(aa\). The two copies in a heterozygous genotype are conventionally written without indicating an order: \(Aa\) and \(aA\) name the same genotype at this locus. The genotypes \(AA\) and \(aa\) are homozygous because their two alleles match; \(Aa\) is heterozygous because they differ.
These labels are bookkeeping, not claims about biological effect. The symbols \(A\) and \(a\) do not by themselves establish which allele is common, harmful, dominant, or functional. Such interpretations require information beyond the notation. A genotype at one locus also does not describe an individual's full genetic makeup: it records only the two alleles at the specified location.
Worked Example: Naming a Genotype at a Locus
Suppose a hypothetical diploid plant is examined at a specified locus where the two allele labels are \(R\) and \(r\). Its two copies at that location are recorded as one \(R\) allele and one \(r\) allele. State its genotype and classify it.
The genotype is \(Rr\), because the individual carries one copy of each allele at the locus. Since the copies differ, this is a heterozygous genotype. If both copies were \(R\), the genotype would be \(RR\), which is homozygous; if both were \(r\), it would be \(rr\), also homozygous.
The description applies only to this locus. It does not imply that the individual carries only two alleles across its genome, nor does it tell us the frequency or effect of either allele in a population.
Allele Frequencies and Genotype Frequencies
A frequency is a proportion relative to a specified collection. The denominator must be clear: allele frequency uses allele copies at a locus, whereas genotype frequency uses individuals at that locus. For a diploid population of \(N\) individuals, there are \(2N\) copies of the locus, assuming each individual has two observed copies and there is no missing genotype information.
For two alleles \(A\) and \(a\), denote their allele frequencies by \(p_A\) and \(p_a\). Denote the genotype frequencies by \(f_{AA}\), \(f_{Aa}\), and \(f_{aa}\). The subscripts identify the allele or genotype being counted. The \(p\) notation here refers to an allele frequency; the \(f\) notation refers to a genotype frequency.
The first equation sums over allele types; the second sums over genotype types. They have different denominators and describe different distributions. For example, \(p_A=0.6\) means that 60% of the allele copies are \(A\). It does not mean that 60% of individuals have genotype \(AA\), nor does it specify what proportion have genotype \(Aa\).
Worked Example: Calculating an Allele Frequency From Allele Copies
In a hypothetical diploid population, a complete tally at one locus records 30 copies of allele \(B\) and 10 copies of allele \(b\). Find both allele frequencies.
There are \(30+10=40\) allele copies in the tally. Thus the frequency of \(B\) is \(p_B=30/40=0.750\), and the frequency of \(b\) is \(p_b=10/40=0.250\).
The check is \(0.750+0.250=1.000\). These values describe the distribution of allele copies in this hypothetical population. They do not, on their own, determine the proportions of \(BB\), \(Bb\), or \(bb\) individuals.
Worked Example: Calculating Genotype Frequencies
A hypothetical survey records genotypes at one locus for 20 individuals: 8 are \(CC\), 9 are \(Cc\), and 3 are \(cc\). Find the three genotype frequencies.
The denominator is the number of individuals, 20. Therefore, \(f_{CC}=8/20=0.400\), \(f_{Cc}=9/20=0.450\), and \(f_{cc}=3/20=0.150\).
Their sum is \(0.400+0.450+0.150=1.000\), as required because every surveyed individual has exactly one of the three listed genotypes at this locus. The heterozygote frequency is 0.450, meaning that 45% of these 20 individuals were \(Cc\). It is not a probability statement about every individual in every population.
Genotype Probabilities for an Individual
A probability refers to an event under a specified probability model or sampling scheme. Let \(G\) be the genotype of an individual selected according to that scheme. For a two-allele locus, genotype events can be written \(G=AA\), \(G=Aa\), and \(G=aa\). If these are the only possibilities, their probabilities satisfy \(P(G=AA)+P(G=Aa)+P(G=aa)=1\).
The probability \(P(G=Aa)\) can describe the chance that a randomly selected individual has genotype \(Aa\), provided the population and selection procedure are specified. It is not automatically a probability about a particular named individual. If a particular person's genotype is known, it is a fixed recorded state; if it is unknown, a probability statement requires a model and the available information. A probability based on uncertainty from a genetic test, for example, would need to be defined using that test information and is not simply the population genotype frequency.
Worked Example: Probability for a Randomly Selected Individual
Consider a hypothetical population model in which a randomly selected individual has probabilities \(P(G=DD)=0.50\), \(P(G=Dd)=0.30\), and \(P(G=dd)=0.20\). Find the probability that the selected individual is heterozygous.
The heterozygous event is \(G=Dd\), so its probability is \(P(G=Dd)=0.30\). The probabilities form a valid distribution because \(0.50+0.30+0.20=1.00\).
Under the stated model and random-selection scheme, the probability is 0.30, or 30%. This describes repeated selections under that scheme; it does not mean that a particular selected individual is “30% heterozygous.” An individual either has genotype \(Dd\) or does not.
When a Probability Equals a Population Frequency
A genotype frequency is a descriptive proportion for a defined population. A genotype probability is a probability assigned under a model or selection procedure. These quantities coincide in a useful special case: select one individual uniformly at random from a finite population, and define the genotype probability from that selection. Then the chance of selecting an individual with a given genotype equals the fraction of the population with that genotype.
The equality follows because each population member has the same chance of being selected. If \(N\) individuals are present and \(n_g\) have genotype \(g\), then each person has selection probability \(1/N\). The event \(G=g\) includes exactly \(n_g\) people, so its probability is the sum of their selection probabilities, \(n_g(1/N)=n_g/N=f_g\).
The equality need not hold when the selection scheme is not uniform or the target differs from the defined population. For example, a sample deliberately enriched for one clinical group does not necessarily have genotype proportions equal to those in the source population. The sampling frame and the population being described therefore belong in any careful interpretation.
Worked Example: A Population Proportion Versus a Selection Probability
A hypothetical finite population contains 50 individuals, of whom 15 have genotype \(EE\). First find the genotype frequency in this population. Then state the probability of \(EE\) for a uniform random selection of one individual from the same population.
The population genotype frequency is \(f_{EE}=15/50=0.300\), or 30%. Under uniform random selection, each of the 50 people is equally likely to be chosen, so \(P(G=EE)=15/50=0.300\).
The numerical values match because the probability model selects uniformly from precisely the population whose proportion was calculated. If the selection favored some individuals over others, this equality would not follow from the population frequency alone.
A Short Check on the Frequency Definitions
The sum-to-one properties are consequences of exhaustive, mutually exclusive categories. For allele copies, every copy at the locus is exactly one of the listed alleles. If \(M\) copies are tallied and \(M_A\) are \(A\), while \(M_a\) are \(a\), then \(M_A+M_a=M\). Dividing both sides by \(M\) gives
For genotypes, every individual with complete data has exactly one genotype from the exhaustive set. If \(N_{AA}\), \(N_{Aa}\), and \(N_{aa}\) are the corresponding counts, then their sum is \(N\). Dividing by \(N\) gives
These checks depend on the stated categories being exhaustive and the denominators being appropriate. If genotypes are missing, for instance, a frequency calculated among individuals with recorded genotypes uses that observed subset as its denominator; it is not automatically the frequency among all individuals in the target population.
Common Mistakes and Why the Distinction Matters
- Using the wrong denominator. Allele frequency is a fraction of allele copies; genotype frequency is a fraction of individuals. State which one is being reported.
- Equating \(p_A\) with \(f_{AA}\). These are different quantities. \(p_A\) counts copies of allele \(A\); \(f_{AA}\) counts individuals with two \(A\) copies.
- Treating an allele frequency as enough to determine genotype frequencies. Knowing the proportions of allele copies does not, by itself, specify how those copies are paired into individuals.
- Calling a population proportion an individual probability without qualification. The equality holds for uniform random selection from the same finite population, but a different sampling scheme can give a different probability.
- Reading biological meaning into allele labels. \(A\) and \(a\) are names. Their labels alone do not establish dominance, function, or clinical importance.
Check Your Understanding
Answer each question using the distinction between allele copies, individual genotypes, and probability models.
- At a locus with alleles \(M\) and \(m\), what genotype does a diploid individual carry if its two copies are \(M\) and \(m\), and is it homozygous or heterozygous?
- A tally contains 18 copies of allele \(K\) and 12 copies of allele \(k\). What are the two allele frequencies, and what denominator did you use?
- In a group of 40 individuals, 10 have genotype \(LL\). What is the genotype frequency of \(LL\)? Explain why the denominator is not the number of allele copies.
- When does the probability that a selected individual has genotype \(Aa\) equal the genotype frequency \(f_{Aa}\) in a finite population?
- Why does knowing \(p_A\) alone not tell you the genotype frequency \(f_{AA}\)?