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Solutions
8.1.
Answers will vary considerably, but students might think about subjective probability as a degree of
belief, uncertainty in one’s mind, or a willingness to bet or accept lotteries. They may appropriately
contrast the subjective interpretation of probability with a frequency interpretation.
8.2.
The model under discussion relates a number of financial ratios to the probability of default. First,
there is a good deal of subjective judgment involved in deciding which financial ratios to include in such a
model. Second, in using past data your friend has made the implicit assumption that no fundamental
changes in causes of default will occur, or that the data from the past are appropriate for understanding
which firms may default in the future. A bank officer using this model is making an implicit subjective
judgment that the model and data used are adequate for estimating the probability of default of the
particular firms that have applied for loans. Finally, the bank officer implicitly judges that your friend has
done a good job!
8.3.
Assessing a discrete probability requires only one judgment. Assessing a continuous probability
distribution can require many subjective judgments of interval probabilities, cumulative probabilities, or
fractiles in order to sketch out the CDF. Even so, the fundamental probability assessments required in the
continuous case are essentially the same as in the discrete case.
8.4.
Answers will, of course, vary a lot. As a motivation for careful quantitative modeling of probability, it
is instructive to collect responses from a number of people in the class and show the ranges of their
responses. Thus, it is clear that different people interpret these verbal phrases in different ways.
Answers can be checked for consistency, as well. In particular, a > 0.5, g > 0.5, l < j, e < j, p < m < i,
and o < f < k.
8.5.
Answers will vary here, too, but many students will decompose the assessment into how well they will
do on homework (for which they have a good deal of information) and how well they will do on a final
exam or project.
Course
Grade
Final
Exam
Homework
8.6.
The students’ assessments may vary considerably. However, they should be reasonable and indicate in
some way that some effort went into the assessment process. Answers should include some discussion of
thought processes that led to the different assessments.
8.7.
It is possible to assess probabilities regarding one’s own performance, but such assessments are
complicated because the outcome depends on effort expended. For example, an individual might assess a
relatively high probability for an A in a course, thus creating something of a personal commitment to work
hard in the course.
8.8.