School of Engineering
Jonsson Engineering Center
ENGR 2600
Modeling and Analysis of Uncertainty
Instructor: Dr. Uwe Kruger (BME, JEC7048)
In-class examples: Introduction
Problem 1 – Tree diagram I
A teaching evaluation has 15 questions with 5 possible answers each. How many different ways there are to fill
out the form, i.e. what is the size of the sample space?
Problem 2 – Tree diagram II
An automobile manufacturer provides vehicles equipped with selected options. Each vehicle is ordered with or
without automatic transmission, with or without air-conditioning, with one of three choices of stereo systems and
with one of four exterior colors. What is the sample space of all possible vehicle types and what is the size of the
sample space?
Problem 3 – Tree diagram and frequency table
Cooking oil is produced in two varieties, mono and polyunsaturated. Two common sources of cooking oil are
corn and canola. The following table shows the number of bottles of these oils in a supermarked.
Frequency table
Type of oil
type of unsaturation
𝐴′ mono
𝐴 poly
𝐵 : canola
23
17
𝐵′ : corn
45
15
Suppose a bottle of oil is selected at random. Let 𝐴 denote the event that the bottle has polyunsaturated oil and
let 𝐵 denote the event that the bottle contains canola oil. Determine 𝑃(𝐴′ ∪ 𝐵).
Problem 4 – Axioms of Probability I
If 𝑃(𝐴) = 0.3, 𝑃(𝐵) = 0.2 and 𝑃(𝐴 ∩ 𝐵) = 0.1, determine 𝑃(𝐴′), 𝑃(𝐴′ ∩ 𝐵′) and 𝑃(𝐴 ∪ 𝐵′).
Problem 5 – Axioms of Probability II
A computer system uses passwords that are six characters and each character is one of the 26 letters (a – z) or 10
integers (0 – 9). Uppercase letter are not used. Let 𝐴 denote the event that a password begins with a vowel (a, e,
i, o, u) and let 𝐵 denote the event that a password ends with an even number (0, 2, 4, 6, 8). Suppose a hacker
selects a password at random.
1. What is the probability that the hacker gets your password if you are one of the users of the system?
2. Determine 𝑃(𝐴), 𝑃(𝐵), 𝑃(𝐴 ∩ 𝐵) and 𝑃(𝐴 ∪ 𝐵).