01
Venn diagrams - set rules and Kolmogorov additivity
Suppose we know three probabilities of events:
, , and . Calculate:
, , , , and .
Solution
04
(1)
is computed by directly applying the inclusion-exclusion principle.
(2)
(3)
(4) We can express
as . Therefore,
(5)
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02
Inclusion-exclusion reasoning
Your friend says: “according to my calculations, the probability of
is and the probability of is , but the probability of and both happening is only .” You tell your friend they don’t understand probability. Why?
Solution
05
(1) State the inclusion-exclusion principle.
(2) Examine possibilities based on given values.
Given that
is and is , we have that . Since
, we know that . Therefore,
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03
Inclusion-exclusion reasoning
Suppose
and . Show that .
Solution
17
(1) State the inclusion-exclusion principle.
(2) Examine the maximum value of
. We know that
. Given that
and , .
(3) Examine the minimum value of
. The minimum value of
is the maximum of the individual values.
. Therefore,
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04
At least two heads from three flips
A coin is flipped three times.
What is the probability that at least two heads appear?
Solution
07
(1) Describe the sample size of this experiment.
(2) Find the probability that at least two heads appear.
The sequences of flips that contain at least two heads are
, , , . We know that
, thus Link to original
05
Which college after Greenville High?
Suppose every student from Greenville High goes on to take classes at College A, B, or C or some combination of those. 30% of students take classes at College A, 40% at B, and 50% at C. Assume that no student takes classes at both College B and College C.
(a) Is it true that everyone who takes classes at A also takes classes at C?
(b) Find the probability that a student will not take classes at B and not take classes at C.
06
Researcher’s degree
Of 1000 researchers at a research laboratory, 375 have a degree in mathematics, 450 have a degree in computer science, and 150 of the researchers have a degree in both fields. One researcher’s name is selected at random.
(a) What is the probability that the researcher has a degree in mathematics, but not in computer science?
(b) What is the probability that the researcher has no degree in either mathematics or computer science?
Solution
06
(a)
(b)
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