Decision Science II · ORL 30306 Exercise Collection
Decision Science II
ORL 30306 — Wageningen University
Exercise Collection
Bayesian Updating · Expected Utility & Certainty Equivalents · Value of
Information · Portfolio Theory · Stochastic Dominance
Part Topic Exercises
1 Bayesian updating (DA2/DA3) 1.1 – 1.5
2 Expected utility & certainty equivalents (Lecture 7) 2.1 – 2.3
3 Utility application: fertilizer decision (DA3) 3.1
4 Eliciting your own utility function (self-test) 4.1
5 Bayesian decision analysis: cattle purchase (DA12) 5.1
6 Portfolio theory (DA10) 6.1 – 6.2
7 Stochastic dominance (DA10) 7.1
Page 1
, Decision Science II · ORL 30306 Exercise Collection
Part 1 — Bayesian Updating
Bayes' theorem lets us update a prior degree of belief P(H) into a posterior P(H | E) once
evidence E is observed:
P(H | E) = P(E | H) · P(H) / P(E), with P(E) = Σ P(E | H ) · P(H )
i i
Exercise 1.1 — Medical test
Consider a rare and lethal disease that affects 1 out of every 1000 people. A test for this
disease is extremely accurate: if you have the disease, the test comes back positive 99% of
the time; if you do not have the disease, it comes back negative 98% of the time.
a) Suppose you test positive. What is the probability that you actually have the disease,
P(D+ | T+)?
Exercise 1.2 — Iterated testing
A positive test that implies only a small chance of disease seems useless — unless you test
again. You take the test a second time, now believing P(Disease) = 0.047 (all other numbers
unchanged).
a) What is P(D+ | T+) after a second positive test?
b) And after a third positive test?
Exercise 1.3 — Spam filter
A classic application of Bayesian updating in artificial intelligence is the Bayesian spam filter,
which helps e-mail services decide whether a message is spam based on certain words.
a) Given that an e-mail contains the word 'free', what is the updated probability that it is
spam, assuming 20% of all e-mails are spam, 60% of spam e-mails contain the word 'free',
and 5% of non-spam e-mails contain the word 'free'?
Exercise 1.4 — John Hinckley's trial
Approximately 1.5% of the US population suffers from schizophrenia. In 1982 John Hinckley
was on trial, accused of the attempted assassination of President Reagan. During the trial an
expert witness told the court that CAT scans of individuals diagnosed with schizophrenia
showed brain atrophy in 30% of cases, against only 2% for 'normal' people. Hinckley's defense
wanted to introduce his CAT scan — which showed brain atrophy — as evidence that he
suffered from this mental illness.
a) What is the probability that Hinckley was schizophrenic given that he had brain atrophy,
P(Sch+ | Atr+)?
b) Does the scan meaningfully strengthen the defense's case? Compare posterior with
prior.
Exercise 1.5 — Searching for a lost aircraft over the ocean
An aircraft disappears over the ocean and search teams must locate it. Based on radar data,
satellite signals and ocean-drift models, investigators divide the potential crash area into four
regions (A, B, C, D) with initial estimates:
Region Prior probability Remark
A 0.50 most probable, near last known location
B 0.30
Page 2
Decision Science II
ORL 30306 — Wageningen University
Exercise Collection
Bayesian Updating · Expected Utility & Certainty Equivalents · Value of
Information · Portfolio Theory · Stochastic Dominance
Part Topic Exercises
1 Bayesian updating (DA2/DA3) 1.1 – 1.5
2 Expected utility & certainty equivalents (Lecture 7) 2.1 – 2.3
3 Utility application: fertilizer decision (DA3) 3.1
4 Eliciting your own utility function (self-test) 4.1
5 Bayesian decision analysis: cattle purchase (DA12) 5.1
6 Portfolio theory (DA10) 6.1 – 6.2
7 Stochastic dominance (DA10) 7.1
Page 1
, Decision Science II · ORL 30306 Exercise Collection
Part 1 — Bayesian Updating
Bayes' theorem lets us update a prior degree of belief P(H) into a posterior P(H | E) once
evidence E is observed:
P(H | E) = P(E | H) · P(H) / P(E), with P(E) = Σ P(E | H ) · P(H )
i i
Exercise 1.1 — Medical test
Consider a rare and lethal disease that affects 1 out of every 1000 people. A test for this
disease is extremely accurate: if you have the disease, the test comes back positive 99% of
the time; if you do not have the disease, it comes back negative 98% of the time.
a) Suppose you test positive. What is the probability that you actually have the disease,
P(D+ | T+)?
Exercise 1.2 — Iterated testing
A positive test that implies only a small chance of disease seems useless — unless you test
again. You take the test a second time, now believing P(Disease) = 0.047 (all other numbers
unchanged).
a) What is P(D+ | T+) after a second positive test?
b) And after a third positive test?
Exercise 1.3 — Spam filter
A classic application of Bayesian updating in artificial intelligence is the Bayesian spam filter,
which helps e-mail services decide whether a message is spam based on certain words.
a) Given that an e-mail contains the word 'free', what is the updated probability that it is
spam, assuming 20% of all e-mails are spam, 60% of spam e-mails contain the word 'free',
and 5% of non-spam e-mails contain the word 'free'?
Exercise 1.4 — John Hinckley's trial
Approximately 1.5% of the US population suffers from schizophrenia. In 1982 John Hinckley
was on trial, accused of the attempted assassination of President Reagan. During the trial an
expert witness told the court that CAT scans of individuals diagnosed with schizophrenia
showed brain atrophy in 30% of cases, against only 2% for 'normal' people. Hinckley's defense
wanted to introduce his CAT scan — which showed brain atrophy — as evidence that he
suffered from this mental illness.
a) What is the probability that Hinckley was schizophrenic given that he had brain atrophy,
P(Sch+ | Atr+)?
b) Does the scan meaningfully strengthen the defense's case? Compare posterior with
prior.
Exercise 1.5 — Searching for a lost aircraft over the ocean
An aircraft disappears over the ocean and search teams must locate it. Based on radar data,
satellite signals and ocean-drift models, investigators divide the potential crash area into four
regions (A, B, C, D) with initial estimates:
Region Prior probability Remark
A 0.50 most probable, near last known location
B 0.30
Page 2
