First course in probability and statistics
Department of mathematics and systems analysis
Aalto University
2A
J Tölle & S Moradi
Spring 2017
Exercise 2A
Basic concepts of probability
Basic rules of probability
Symmetric probability and combinatorics
Axioms of Probability
Law of total probability and Bayes’ formula
Class exercises
2A1 We roll two symmetric dice with numbers 1, 2, 3, 4, 5, and 6 on the lateral faces. Let
us consider the sum of outcomes
z = x + y,
where x = outcome of the first roll and y = outcome of the second roll. We define events
A = {The sum is 1},
B = {The sum is 11},
C = {The outcome of the first roll is 2},
D = {The outcome of the first roll is 5}.
(a) What is the sample space of the sum of outcomes z = x + y.
(b) What is the probability of event A.
(c) What is the probability of event B.
(d) What is the conditional probability of event B if C has already occurred?
(e) What is the conditional probability of event B if D has already occurred?
2A2 There are 10 000 students in a university. The table below illustrates the age and sex
distribution of the students. What are the probabilities of the following events?
(a) Randomly chosen student is a female.
(b) Randomly chosen student is a male, given that he is 25-34 years old.
(c) Randomly chosen student is male or 25-34 years old.
Age 14-17 18-24 25-34 ≥ 35
Male
50
2 500 1 000 400
Female 150 3 500 1 500 900
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First course in probability and statistics
Department of mathematics and systems analysis
Aalto University
J Tölle & S Moradi
Spring 2017
Exercise 2A
Homework
2A3 The table below illustrates the composition of 101th US congress (chosen 1988). Congressmen are classified into three classes according to the political orientation and the
time as a congressman. The table shows the probabilities obtained by treating political
orientation and the time as a congressman separately (marginal probabilities). Fill the
empty cells of the table under the assumption that the political orientation and the time
as a congressman are independent.
Time as a congressman
< 2 years
2-9 years
≥ 10 years
Total
Political orientation
Democrat Republican
Total
0.090
0.478
0.432
0.614
0.386
2A4 (a) We want to know the reliability of a polygraph machine based on the following
information: A person who is lying, is correctly classified as a liar with probability
0.9. On the other hand a person who does not lie, will be incorrectly classified as
a liar with probability 0.05. We use the polygraph machine in the group where we
know that 1% is lying. What is the probability that the person classified as a liar
is actually honest?
(b) A data transmission system transfers the bits 0 and 1. 60% of transmitted bits are
zeros and 40% are ones. However, some random errors occur in the system and
some of the zeros change into ones and vice versa. The probability that 0 comes
through as 0 is 0.95 and the probability that 1 comes through as 1 is 0.9. What is
the probability of event ”the submitted bit is 0 given that the received bit is 0”?
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