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Solutions
7.1. We often have to make decisions in the face of uncertainty. Probability is a formal way to cope with
and model that uncertainty.
7.2. An uncertain quantity or random variable is an event that is uncertain and has a quantitative outcome
(time, age, $, temperature, weight, . . . ). Often a non-quantitative event can be the basis for defining an
uncertain quantity; specific non-quantitative outcomes (colors, names, categories) correspond to
quantitative outcomes of the uncertain quantity (light wavelength, number of letters, classification number).
Uncertain quantities are important in decision analysis because they permit us to build models that may be
subjected to quantitative analysis.
7.3. P(A and B) = 0.12 P(B
— ) = 0.35
P(A and B
— ) = 0.29 P(B | A) = 0.12
0.41 = 0.293
P(A) = 0.41 P(A | B) = 0.12
0.65 = 0.185
P(B) = 0.65 P(A
– | B
— ) = 0.06
0.35 = 0.171
7.4. P(A or B) = P(A and B) + P(A and B
— ) + P(A
– and B)
= 0.12 + 0.53 + 0.29 = 0.94
or P(A or B) = P(A) + P(B) - P(A and B)
= 0.41 + 0.65 - 0.12 = 0.94
or P(A or B) = 1 - P(A
– and B
— ) = 1 - 0.06 = 0.94
7.5.
A
B
A and B
A and B A and B
From the diagram, it is clear that
P(A) = P(A and B) + P(A and B
— )
and
P(B) = P(A and B) + P(A
– and B).
But P(A or B) clearly equals P(A and B) + P(A and B
— ) + P(A
– and B) because of property 2. Thus,
