V63.0233, Theory of Probability Name:
Worksheet for Sections 2.1–2.3 : Sample Spaces and the Axioms of Probability July 1, 2009
1. A small taxicab company owns four cabs. Let X be the sample space of pairs (x, y), repre-
senting the outcome that x cabs are in operative condition and y of those cabs are out on a call.
Let K be the event that at least two cabs are out on a call, L the event that exactly one of the
operative cabs is not out on a call, and M the event that only one of the cabs is operative.
(a) Draw the sample space using points in a Cartesian plane.
(b) List and indicate on your diagram the event K.
(c) List and indicate on your diagram the event L.
(d) List and indicate on your diagram the event M .
2. Continuing the above problem, express in words what events are represented by the following
sets of points:
(a) N = {(1, 1), (2, 1), (3, 1), (4, 1)}
(b) O = {(2, 0), (3, 0), (4, 0), (4, 1), (4, 2)}
(c) P = {(0, 0), (1, 0), (1, 1)}
3. Continuing the two previous problems, which of the following events are mutually exclusive?
(a) K and M (d) L and P
(b) K and O (e) O and P
(c) M and N (f) K and N
4. Referring to the sample space of Problems 1 and 2, list the points which constitute the
following events, and express them in words:
(i) K (iv) M ∩ P (vii) O ∩ P
(ii) O (v) K ∪ N (viii) M ∪ P
(iii) L ∪ N (vi) L ∩ M (ix) O ∩ L
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5. Analyzing business conditions in general, four government officials make the following claims:
The first claims that the probabilities for unemployment to go up, remain unchanged, or go down
are, respectively, 0.51, 0.33, and 0.12, the second claims that the respective probabilities are 0.55,
0.49, and −0.04, the third claims that the respective probabilities are 0.52, 0.38, and 0.10, and the
fourth claims the respective probabilities are 0.48, 0.34, and 0.21. Comment on these claims.
6. If a certain mathematics professor asks a question of one of his students (chosen at random
from a very large class), the probabilities that his student will have received an A, B, C, D, or F
in the last examination are 0.09, 0.23, 0.36, 0.18, and 0.14. What are the probabilities that the
student received
(i) at least a C?
(ii) at least a D?
(iii) a B, C, or D?
(iv) at most a C?
Assume that each student in the class has taken the examination.
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