Premed · Premed · Genetics
Lecture 2: Probability and Pedigree Analysis
Genetics
Learning Objectives
By the end of this lecture, students will be able to:
- Apply the product rule and sum rule of probability to genetic crosses
- Use the binomial expansion to calculate probabilities of specific offspring distributions
- Interpret standard pedigree symbols and conventions
- Determine the mode of inheritance from pedigree analysis (autosomal dominant, autosomal recessive, X-linked dominant, X-linked recessive)
- Calculate carrier probabilities from pedigree data
- Apply Bayesian analysis to refine genetic risk estimates
Lecture Content
I. Probability Rules in Genetics
Probability (P) ranges from 0, representing an impossible event, to 1, representing a certain event. Three probability rules are especially important in genetics.
The multiplication rule (product rule) states that the probability of two independent events both occurring equals the product of their individual probabilities: P(A and B) = P(A) x P(B). For example, the probability that a heterozygous cross (Aa x Aa) produces a homozygous recessive child is 1/4, and the probability of two such children in a row is 1/4 x 1/4 = 1/16.
The addition rule (sum rule) states that the probability of either of two mutually exclusive events occurring equals the sum of their individual probabilities: P(A or B) = P(A) + P(B). For instance, the probability of an Aa x Aa cross producing a heterozygote (Aa) can happen in two ways, either A from the mother and a from the father or a from the mother and A from the father, giving a total probability of 1/4 + 1/4 = 1/2.
Conditional probability describes the probability of an event given that another event has already occurred: P(A|B) = P(A and B) / P(B). As an example, given that an offspring of Aa x Aa shows the dominant phenotype, the probability it is a carrier is calculated as P(Aa | dominant) = (2/4) / (3/4) = 2/3.
II. Binomial Expansion
The binomial expansion is used when asking questions of the form: "What is the probability of getting exactly k successes in n independent trials?" The binomial formula is P(k) = [n! / (k!(n-k)!)] x p^k x q^(n-k), where n is the total number of trials (offspring), k is the number of "successes" (for example, affected offspring), p is the probability of success in a single trial, and q is the probability of failure (1 - p).
Consider a concrete example: for Aa x Aa parents having 5 children, the probability of exactly 2 being affected (aa) is calculated with n = 5, k = 2, p = 1/4, and q = 3/4. This yields P = [5!/(2!3!)] x (1/4)^2 x (3/4)^3 = 10 x 1/16 x 27/64 = 270/1024, which is approximately 0.264. The binomial coefficient [n!/(k!(n-k)!)] counts the number of distinct ways to arrange k successes among n trials.
<image>Panel A: Decision tree diagram showing how the product rule and sum rule apply to a monohybrid cross (Aa x Aa), with branching probabilities leading to each genotypic outcome. Panel B: Worked example of binomial expansion for a family of 4 children from carrier parents (Aa x Aa), showing the probability distribution for 0, 1, 2, 3, and 4 affected children as a bar graph. Panel C: Comparison table summarizing when to use the product rule, sum rule, and binomial formula with genetic examples.</image>
III. Chi-Square Goodness-of-Fit Test
The chi-square goodness-of-fit test is a statistical method used to determine whether observed genetic ratios differ significantly from expected ratios. The formula is: chi-square = sum of [(observed - expected)^2 / expected] for all phenotypic classes.
The procedure involves five steps. First, state the null hypothesis, for example, that the data fit a 3:1 ratio. Second, calculate the expected values from the hypothesis. Third, compute the chi-square statistic. Fourth, determine the degrees of freedom (df = number of classes - 1). Fifth, compare the computed chi-square value to the critical value from a chi-square table at the chosen significance level, usually p = 0.05. If the chi-square exceeds the critical value, the null hypothesis is rejected, meaning the observed data deviate significantly from expected. If the chi-square is less than the critical value, we fail to reject the null hypothesis, meaning the data are consistent with the expected ratio. It is important to recognize that the chi-square test does not prove a hypothesis is correct; it only tests whether data are consistent with expectations.
IV. Pedigree Analysis Fundamentals
A pedigree is a diagram showing the inheritance pattern of a trait through multiple generations of a family. Standard symbols are used to convey information efficiently: a square represents a male and a circle represents a female; a filled symbol indicates an affected individual while an open symbol indicates an unaffected one; a half-filled symbol denotes a known carrier. A horizontal line connecting a male and female indicates a mating, and vertical lines descending from that connection indicate offspring. Generations are labeled with Roman numerals (I, II, III), and individuals within a generation are numbered with Arabic numerals. A diagonal line through a symbol indicates a deceased individual, a double horizontal line between partners indicates a consanguineous mating (between related parents), and a diamond is used when sex is unspecified. The proband, or index case, is marked with an arrow.
<image>Panel A: Reference chart showing all standard pedigree symbols with labels (male, female, affected, carrier, deceased, consanguineous mating, proband arrow, twins). Panel B: Four small example pedigrees side by side, each illustrating a different inheritance pattern: autosomal dominant, autosomal recessive, X-linked dominant, and X-linked recessive, with key distinguishing features highlighted. Panel C: Step-by-step annotation of a three-generation autosomal recessive pedigree showing how to assign genotypes to each individual.</image>
V. Autosomal Dominant Inheritance
In autosomal dominant pedigrees, several characteristic features emerge. The trait appears in every generation without skipping, and affected individuals have at least one affected parent. Unaffected individuals who mate with affected heterozygotes have a 50% chance of having affected offspring, and males and females are affected in approximately equal proportions. Unaffected individuals do not transmit the trait, barring incomplete penetrance, and father-to-son transmission is possible, which distinguishes autosomal dominant from X-linked inheritance. Classic examples include Huntington disease, Marfan syndrome, achondroplasia, and familial hypercholesterolemia.
Several important considerations can complicate this picture. New mutations can cause the trait to appear without an affected parent. Variable expressivity means that severity can differ among affected individuals. Incomplete penetrance means that some individuals carrying the genotype do not express the phenotype at all.
VI. Autosomal Recessive Inheritance
Autosomal recessive pedigrees display a distinct set of features. The trait often "skips" generations, and affected individuals typically have unaffected parents who are both carriers. Two carrier parents have a 25% chance of producing an affected offspring per pregnancy. Males and females are affected in approximately equal proportions, and consanguinity increases the likelihood of affected offspring because related parents are more likely to carry the same recessive allele. The trait may appear to "come from nowhere" in families with no known history. Examples include cystic fibrosis, sickle cell disease, phenylketonuria (PKU), and Tay-Sachs disease.
Carrier frequency calculations draw on the Hardy-Weinberg equilibrium. If the disease frequency equals q^2, then the allele frequency q can be determined as the square root of q^2, and the carrier frequency (2pq) can be calculated accordingly.
VII. X-Linked Inheritance
X-linked recessive conditions affect males far more frequently than females because males are hemizygous for the X chromosome and need only one copy of the recessive allele to be affected. Affected males inherit the allele from their carrier mothers, and no male-to-male transmission occurs because fathers pass the Y chromosome to their sons. Carrier females are usually unaffected but may show mild expression due to X-inactivation, and all daughters of an affected male are obligate carriers. Examples include hemophilia A, Duchenne muscular dystrophy, and red-green color blindness.
In X-linked dominant conditions, affected males pass the trait to all of their daughters but to none of their sons. Affected heterozygous females pass the trait to 50% of offspring regardless of sex. Females are affected more frequently than males in the population, and the condition may be lethal in hemizygous males. Examples include Rett syndrome, incontinentia pigmenti, and vitamin D-resistant rickets.
VIII. Bayesian Analysis in Genetics
Bayesian analysis is used to update the probability of a genotype based on additional evidence, such as having unaffected children. The analysis involves four components: the prior probability, which is the initial probability based on pedigree position; the conditional probability, which is the probability of the observed evidence given each hypothesis; the joint probability, calculated as the product of prior and conditional; and the posterior probability, which is the joint probability for one hypothesis divided by the sum of all joint probabilities.
Consider an example: a woman whose brother has cystic fibrosis (autosomal recessive) has a 2/3 prior probability of being a carrier. She has 3 unaffected children with a non-carrier partner. If she is a carrier, the probability of 3 unaffected children is (3/4)^3 = 27/64. If she is not a carrier, the probability of 3 unaffected children is 1. The posterior probability of her being a carrier is then (2/3 x 27/64) / [(2/3 x 27/64) + (1/3 x 1)]. This approach allows for more precise genetic counseling by incorporating all available family data into risk estimates.
<image>Panel A: A detailed three-generation pedigree for an autosomal recessive condition with genotypes assigned to each individual (AA, Aa, or aa), including carrier status indicated by half-filled symbols. Panel B: Bayesian analysis table with columns for "Carrier" and "Non-carrier" hypotheses, rows for prior probability, conditional probability, joint probability, and posterior probability, filled in with a worked example. Panel C: A pedigree demonstrating X-linked recessive inheritance showing the characteristic pattern of affected males, carrier females, and absence of father-to-son transmission.</image>
