18.5 Problems
Problem 1. Let’s roll two six-sided dice! Describe the event space, σ-algebra, and the corresponding probabilities for this experiment.
Problem 2. Let Ω = [0,1], and the corresponding σ-algebra be the generated algebra
![( ) Σ = σ {(a,b] : 0 ≤ a <b ≤ 1} .](https://static.packt-cdn.com/products/9781837027873/graphics/media/file1725.png)
Show that the following events are members of Σ:
(a) S1 = {x} for all x ∈ [0,1].
(b) S2 = ∪i=1n(ai,bi). (Show that this is also true when the intervals […] are replaced with open and half-open versions (…),(…],[…).)
(c) S3 = [0,1] ∩ℚ. (That is, the set of rational numbers in [0,1].)
(d) S4 = [0,1] ∩(ℝ ∖ℚ). (That is, the set of irrational numbers in [0,1].)
Problem 3. Let’s roll two six-sided dice. What is the probability that
(a) Both rolls are odd numbers?
(b) At least one of them is an odd number?
(c) None of them are odd numbers?
Problem 4. Let Ω = ℝ2 be the event space, where we define the open disks
=
and the open...
