Stat 134: Section 2
Tianyu Guo Email: tianyu_guo@berkeley.edu
Conceptual Review
Let the experiment consist of a roll of a pair of dice (as in the games
of Monopoly or craps). We assume that the dice can be distinguished
from each other, for example that one of them is blue and the other
one is red.
1. Give the sample space, and the probability for each outcome.
2. What is the probability of getting the same number?
3. What is the probability of getting one number being two times the
other one.
4. What is the probability of getting the first number being even and
second one being odd?
5. What is the probability that the sum of the two numbers is 6?
6. What is the probability of the events in parts 3 and 4 both happen?
7. What is the probability of the events in parts 3 and 5 both happen?
8. What is the probability of the events in parts 4 and 5 both happen?
9. What is the probability of the events in parts 3, 4, and 5 all happen?
10. How would the answers change if two dice are indistinguishable?
stat 134: section 2
HW1-3
Let A1 , . . . , A5 be events on the same probability space. Express the
following events using the usual set operations (∁, ∪, ∩) and the
events Ak , 1 ≤ k ≤ 5.
1. All events occur.
2. At least one event occurs.
3. At least two of them occur.
4. A1 occurs, but A2 doesn’t.
5. A1 occurs, but A2 and A3 don’t occur together.
6. A1 occurs, or A2 and A3 don’t occur together.
7. At least two of the five events occur, but the five events don’t
occur at the same time.
HW1-5
A seemingly natural construction for “uniform” distribution on natural numbers: A set A ⊂ {1, 2, . . .} is said to have asymptotic density θ
if
lim |{1, . . . , n} ∩ A|/n → θ.
(1)
n→∞
Let A be the set of even numbers. Then, A ∈ A with θ = 12 . Let
I0 = {1}, Ik = {n : 2k−1 < n ≤ 2k }, for k = 1, . . .. Construct B as
follows.
B=
∞
[
{( I2k ∩ Ac ) ∪ ( I2k+1 ∩ A)}.
k =0
Show that B ∩ A (I wrote B in the previous version, sorry for the
typo) doesn’t have a asymptotic density. This means that asymptotic
density is not a good way to define probability.
2