# Download Lecture 4 - The Department of Statistics and Applied Probability, NUS

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```Probability in genetics
Gregor Mendel and his peas
• Gregor Mendel (1822 – 1884), experimented on
peas
• Mendelian inheritance, single (or very few)
genes controlling certain expressed traits
(dominant and recessive)
dogfoose.wordpress.com/infotoons/
Gregor Mendel and his peas
In Mendel’s experiment, the two copies of the peas of the parental generation are
identical – either two for tallness (DD) or two for dwarfness (dd).
When crossing tall and dwarf plants, each offspring receives one D version (allele,
in genetic terminology) and one d version of the gene.
Since they have both the D and the d allele, their genetic constitution (genotype) is
symbolized by Dd. Since D is dominant, all plants are tall.
When producing further offspring from the first offspring generation, both father and
mother pass on either the D or d allele with the same probability. The possible
outcomes are often summarized in a Punnett square, the probabilities for a given
genotype are calculated by using the multiplication rule for independent events.
Since both DD and Dd
there is a 75% chance that any
offspring will be tall.
Example 1:
In Mendel’s experiment, yellow seeds (G) dominate over green (g) ones,
and round (W) over wrinkled (w) ones. We already know that tall plants (D)
dominate over dwarf (d) ones.
a) Suppose two pea plants that are hybrids (heterozygous) in all three traits
are crossed. What is the chance for a particular offspring to be tall with
yellow, wrinkled seeds?
b) Suppose you are crossing the genotypes Dd / Gg / ww and dd / GG /
Ww. What is the chance that the offspring is dwarf, with yellow, round
seeds?
c) Suppose you are crossing a pea of unknown genotype with a
homozygous recessive one. (This is a pea with genotype dd / gg / ww)
This is called a test-cross in genetics. Suppose that approximately half of
the offspring is dwarf, but all has yellow, wrinkled seeds. Can you find
out the genotype of the parent?
Example 2:
With snapdragons (Antirrhinum majus), homozygous WW genotypes are red,
ww are white, and heterozygous Ww are pink. If we cross a red and a pink
snapdragon, what is the chance that a randomly chosen offspring is pink?
Example 3:
Phenylketonuria, a metabolic genetic disorder in humans, is caused by a
recessive allele, k. If two heterozygous carriers of the allele marry and plan a
family of five children .What is the chance that none of the kids will have the
disease? What is the chance that no more than two will have the disease?
(This is a practical application of probability in biomedical sciences, and is
commonly known as genetic counselling, a formal assessment of the risk
of certain genetic conditions)
Random mating and Hardy-Weinberg
equilibrium
The Hardy-Weinberg equilibrium (HWE) is one of the central concepts in population
genetics. It can be used to predict the genotype frequencies in a population from the
allele frequencies.
Suppose that in a population there are two alleles A and a of a certain gene.
Assume that the relative frequency is p for “A” and q = 1 – p for “a”.
We assume that members of the population mate randomly with respect to this
gene. By this we mean that the choice of a partner for mating is not influenced by
the genotype with respect to this gene. Then both sperms and eggs involved in
mating will contain A with probability p, and a with probability q.
Assuming that egg and sperm cell match at random, an offspring will have genotype
AA with probability p2, Aa with probability 2pq, and aa with probability q2. The
relative frequencies of the genotypes should then be close to these probabilities,
and the population is said to be in a state of HWE.
Example 4:
A random sample of 6129 people is taken and their blood type frequencies
with respect to the M-N classification are recorded.
Calculate the HWE frequencies for the sample.
Practical application in public health
By assuming that a population is in HWE, the frequency for one genotype
can be used to obtain the frequency of all other genotypes. This can be
useful for estimating the frequency of carriers of a disease.
This is also directly relevant to phamacogenetics, and in gauging the chance
of adverse drug reactions in the population.
Example 5:
In the US, the frequency of the recessive metabolic disorder
phenylketonuria (PKU) is about 0.0001. Assuming HWE, can you estimate
the percentage of carriers of PKU in the population.
Example 6:
Genes relating to albinism are denoted by A and a. Only those children who
receive the a gene from both parents will be albino. A person having the
genotype Aa is normal in appearance and, because they can pass on the
trait to their offspring, is called a carrier.
Suppose that a normal couple has two children, exactly one of whom is an
albino. Suppose also the non-albino child mates with a person who is known
to be a carrier for albinism.
(a) What is the probability that their first offspring is an albino?
(b) What is the conditional probability that their second offspring is an albino,
given their first-born is not?
Mendelian genetics
• Mathematical basis to biology
• Relatively easy to detect
• Generally monogenic, coupled with selection
pressures  relatively rare
• Genetic counselling
Genetic architecture of diseases and traits
McCarthy et al. (2008) NRG 9:356-369
Interpreting probability
We have reached the end of the section on probability. Do you think that you
are comfortable with interpreting probability?
Actually the interpretation of probability is extremely subjective, and is
entirely based on the context. Consider the following statements:
“Your chance of getting diabetes before the age of 60 is 60%.”
“You scored 60% for your PSLE mathematics examination.”
“The predictive ability of genes on the chance of diabetes onset is currently
around 2%.” (c.f. with random guessing, is the probability 50%? – think