# Worksheet 02

## Sample Space and Events

Assume for any given day, the weather could be either sunny, cloudy, rainy, or snowy. Suppose you are interested in predicting the weather over for the weekend ahead, both Saturday and Sunday. What is the sample space for your weekend weather predictions?

Imagine you own a gas station here in Chico. Suppose you are interested in the zip codes from which you customers arrive. Write down the sample space for customer zip codes and write down a possible event.

## Set Algebra

Suppose \(S = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}\), \(A = \{1, 2, 4, 6\}\), and \(B = \{1, 3, 5\}\). What is

- \(A \cap B\)
- \(A \cup B\)
- \(A^c\)

Let \(A_k = [0, 1/k)\). Write the following sets as intervals

- \(\cap_{n=1}^5 A_n\)
- \(\cup_{n=1}^{10} A_n\)

Let \(A\) be any set. Find the set \(A \cap A^c\).

# Axioms of Probability

Suppose \(S = \{ 1, 2, 3, 4, 5, 6, 7, 8 \}\) such that \(\mathbb{P}[\{s\}] = 1/8\) for \(1 \leq s \leq 8\).

- What is \(\mathbb{P}[\{1, 2\}]\)?
- What is \(\mathbb{P}[\{1, 2, 3\}]\)?

Suppose \(S = \{1, 2, 3\}\) such that \(\mathbb{P}[\{1\}] = 1/2\), and \(\mathbb{P}[\{1, 2\}] = 2/3\). What must the value of \(\mathbb{P}[\{2\}]\) be?

# Properties of Probability/Distributions

- Suppose \(S = \{ 1, 2, \ldots, 100\}\). If \(\mathbb{P}[\{1\}] = 0.1\),
- what is \(\mathbb{P}[\{2, 3, 4, \ldots, 100\}]\)?
- what is the smallest possible value of \(\mathbb{P}[\{1, 2, 3\}]\)?

- Suppose that a students arrives late 10% of the time, leaves early 20% of the time, and both arrives late and leaves early 5% of the time. What is the probability that on a given day that student will either arrive late or leave early (or both)?

# Uniform Distribution

Most California license plates contain 7 characters in the following order: 1 digit, 3 letters (uppercase only), and then 3 digits. If each character is randomly and uniformly selected, within its position, what is the probability

- any particular license plate is selected?
- all three letters are non-repeating vowels and the remaining digits can be any number?