Statistical evidence evaluation
Exercises 1
1. On a parking ground there are five cars parked. The cars are
A blue Volvo with manual gearbox
A red Volvo with manual gearbox
A red Volvo with automatic gearbox
A red Hyundai with manual gearbox
A red Mercedes with automatic gearbox
Suppose you choose a car at random from the five cars.
Let
A = The chosen car is a Volvo
B = The chosen car has an automatic gearbox
C = The car is red
Are A and B independent events? Are they conditionally independent given C?
2. Dye on banknotes (from first lecture)
Assume a method for detecting a certain kind of dye on banknotes is such that
⋅ It gives a positive result (detection) in 99 % of the cases when the dye is present, i.e.
the proportion of false negatives is 1%
⋅ It gives a negative result in 98 % of the cases when the dye is absent, i.e. the
proportion of false positives is 2%.
The presence of dye is rare: prevalence is about 0.1 %.
If the method gives a positive result, what is the conditional probability that the dye is
present?
The answer to the question can be found in the hand-outs from lecture 1. However, how
does the answer depend on the proportion of false negatives and the proportion of false
positives? Make a graphical plot of the conditional probability of interest as a function of
these two proportions. Which of the two proportions is most crucial for the conditional
probability to become satisfactory high?
3. Which are the proper scales for the states of the following variables?
a) Gender
b) Temperature in Fahrenheit degrees
c) Age
d) Acoustic intensity level (given in integral dB values: 45 dB, 89 dB etc.)
4. Suppose that in a village historical records show that in November there will rain on
17 days (on the average). On rainy days, a parking place outside Mr Johnson’s house gets
wet in four out of five days. However, if somebody washes their car up the road of Mr
Johnson’s house the parking place may also be wet. This happens roughly in 50 % of the
cases. We assume that nobody will wash their car on a rainy day in November, and we also
assume there could be no further explanation to why the parking place is wet.
Now, assume that on one November evening it is not raining (for the moment). If you
observe that the parking place outside Mr Johnson’s house is wet, what is the likelihood
that it has rained that day? What can you say about the posterior probability that it has
rained that day?
5. A fair coin should have equal probabilities to land on each side when tossed. We may for
simplicity set this probability to 0.5 (assuming it cannot land end-ways). A specific coin is
assumed to have been manipulated so that the probability of obtaining heads is 0.6 instead
of 0.5. Suppose you toss the coin 5 times and obtain heads in four of these cases. What is
the Bayes factor in testing the hypothesis “The coin is fair” against the hypothesis “The
coin has the probability 0.6 to give heads when tossing” ?
6. Consider the coin in the previous exercises. If the coin is manipulated to give heads more
often than tails any probability of heads larger than 0.5 would do. Assume we have no
particular reason to believe that some probabilities (above 0.5) would be more probable
than others. For the same experiment as in exercise 5, what is the Bayes factor in testing
the hypothesis “The coin is fair” against the hypothesis “The coin is manipulated to give
more heads than tails on the average”.
7. Suppose we suspect that a particular screwdriver was used to break a lock on a door. Some
other information tells us it must have been a screwdriver (unrealistic, but just to simplify).
Suppose further that the particular type of screwdriver would leave precisely those marks
on the lock that have been observed. This type of screwdriver constitutes about 1 % of all
screwdrivers on the market. Among the rest of the screwdrivers on the market it is
appreciated that 0.02 % of them would leave such marks as have been observed. What is
the Bayes factor in testing the hypothesis “The particular screwdriver was used to break the
lock” against the hypothesis “Another screwdriver was used to break the lock”?