ENVM 3503 Semester 1 2003
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1. Horse Race (10 marks)
Let’s assume that there is a race between two horses: Fleetfoot and Dogmeat, and
you want to determine which horse to bet on. Fleetfoot and Dogmeat have raced
against each other on twelve previous occasions, all two-horse races. Of these last
twelve races, Dogmeat won five and Fleetfoot won the other seven. Therefore, all
other things being equal, the probability of Dogmeat winning the next race can be
estimated as 5/12 or 0.417 or 41.7%. However, on three of Dogmeat’s previous five
wins, it had rained before the race. It had rained only once on any of the days that he
lost. On the day of the race in question, it is raining.
Construct a Bayesian network to show the probability of Dogmeat winning the race.
Explain your Bayesian network and how you obtained your answer.
2. Meat Test (10 marks)
Minced meat purchased in the supermarket may be infected with bacteria. On
average, infection occurs once in 600 packages of meat. A test with a positive or
negative result can be used to test for infection. If the meat is clean, the test result
will be negative in 499 out of 500 cases, and if the meat is infected, the test result will
be positive in 499 out of 500 cases.
Construct a Bayesian Network to show the probability of a package of meat being
infected given a positive test result. Explain your Bayesian network and how you
obtained your answer.
3. Flower Breeding (20 marks)
You are a flower breeder. The plant you are breeding can either have red flowers or
white flowers. You know that the colour of a flower depends on the genotype of the
plant. The gene for red flowers (represented by R) is a dominant gene and the gene
for white flowers (represented by r) in a recessive gene. Therefore, a plant with the
genotype RR or Rr has red flowers, while a plant with the genotype rr has white
flowers. Hence, the colour of a plant’s flowers is influenced by its genotype (as
shown in Figure 1) and the probability of a plant having red or white flowers, given its
genotype, is shown in Table 1.
Plant_1_Genotype
RR
Rr
rr
33.3
33.3
33.3
Plant_1_Flower_Colour
Red
White
50.0
50.0
Figure 1: Diagram showing that a plant’s genotype influences the colour of its
flowers.
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Table1: Probability of a plant having red or white flowers given its genotype.
Genotype Probability of flower colour (%)
Red White
RR 100 0
Rr 100 0
rr 0 100
When breeding flowers you know that the genotype (and therefore flower colour) of
an offspring is influenced by the genotype of its parents (as shown in Figure 2).
Plant_1_Genotype
RR
Rr
rr
33.3
33.3
33.3
Plant_2_Genotype
RR
Rr
rr
33.3
33.3
33.3
Offspring_Genotype
RR
Rr
rr
33.3
33.3
33.3
Figure 2: Diagram showing that the genotype of an offspring is influenced by the
genotype of its parents.
You also know that the following parent crosses are possible:
If two plants of genotype RR are mated, then the offspring will always be RR.
If two plants of genotype rr are mated, then the offspring will always be rr.
If a plant of genotype RR is mated with a plant of genotype Rr, then the
offspring will always get an R from one parent and may get an R or an r from
the other parent, which means it could be of genotype RR or Rr.
If a plant of genotype RR is mated with a plant of genotype rr, then the
offspring will always get an R from one parent and will always get an r from
the other parent, which means it will always be of genotype Rr.
If a plant of genotype Rr is mated with a plant of genotype Rr, then the
offspring may get an R or r from one parent and an R or r from the other
parent, which means it could be of genotype RR, Rr or rr.
If a plant of genotype Rr is mated with a plant of genotype rr, then the
offspring may get an R or r from one parent and will always get an r from the
other parent, which means it could be of genotype Rr or rr.
For the above crosses, the probability of offspring being a particular genotype is
given in Table 2.
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Table 2: Probability of offspring genotypes given the genotypes of the parents
Parent Genotypes Probability of Offspring Genotype (%)
Parent 1 Parent 2 RR Rr rr
RR RR 100 0 0
rr rr 0 0 100
RR Rr 50 50 0
RR rr 0 100 0
Rr Rr 25 50 25
Rr rr 0 50 50
Finally, for plants with unknown parent genotypes, you know that the probability of
them being genotype Rr is 50%, while the probability of them being of genotype RR
or rr is 25%.
Now suppose you have two plants. The genotypes of their parents are unknown;
however the flowers of both plants are red. You mate these two plants to produce a
first generation offspring. This offspring is then mated with a third plant, with white
flowers, to produce a second generation offspring.
Construct a Bayesian Network and use it to determine the probability that the second
generation offspring will have red flowers? Explain your Bayesian network and how
you obtained your answer.
4. Horse Stud (20 marks)
You are the manager of a horse stud. A colt called John has been found to suffer
from a life-threatening hereditary disease caused by a recessive gene. The disease
is so serious that John’s parents, Henry and Irene, are taken out of the stud-breeding
program. However, you still need to decide which of the remaining horses in the stud
are likely to carry the disease-causing gene and therefore should be removed from
the breeding program. You look through the stud records to retrace John’s family tree
(Table 1).
Table 1: John’s family tree.
Mare Stallion Foal
Irene Henry John
Dorothy Fred Henry
Gwenn Eric Irene
Jill Jack Fred
Jill Brian Dorothy
Cecily Brian Eric
Cecily Mike Gwenn
You know that in order to have the disease, a horse must carry a double-recessive
gene (aa). John is the only horse in the stud that has the disease so he can be the
only horse of the genotype aa. Therefore, the remaining horses in the breeding
program can either be carriers of the disease causing gene (aA) or pure (AA). You do
some further research and find the probabilities of a foal being diseased (aa), a
carrier (aA) or pure (AA), given the genotype of the father and mother (Table 2).
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Table 2: Probability of a foal being diseased given the genotype of the father and
mother.
Foal
Father Mother aa aA AA
aA aA 25% 50% 25%
aA AA 0% 50% 50%
AA aA 0% 50% 50%
AA AA 0% 0% 100%
We know that John is the only horse with the genotype aa, so for the other horses in
the stud we can remove the probability of them being diseased (aa) and normalise
the remaining probabilities so that they add to 100%. Hence, for the other horses in
the stud, the probability of them being a carrier (aA) or pure (AA) given the genotype
of the father and mother is shown in Table 3.
Table 3: Probability of a foal being a carrier or pure given the genotype of the father
and mother.
Foal
Father Mother aA AA
aA aA 67% 33%
aA AA 50% 50%
AA aA 50% 50%
AA AA 0% 100%
For the horses without a recorded father or mother, we know that the frequency of
occurrence of the recessive gene is 1 in every 100 horses.
Construct a Bayesian Network to help you determine which horses in the stud are
most likely to be carriers of the disease-causing gene and should be culled from the
breeding program. You have one further piece of information to assist your decision –
Fred has previously been tested for the disease-causing gene and he is not a
carrier (he is pure). Which horse(s) will you cull from the breeding program? Explain
your Bayesian network and how you obtained your answer.