Chapter 2: Probability

Meike Niederhausen and Nicky Wakim

2023-09-27

Learning Objectives

Define basic axioms and propositions in probability
Assign probabilities to events, and perform manipulations on probabilities to make calculations easier

Probabilities of equally likely events

Pick an equally likely card, any equally likely card

Example 1

Suppose you have a regular well-shuffled deck of cards. What’s the probability of drawing:

any heart
the queen of hearts
any queen

Let’s break down this probability

If \(S\) is a finite sample space, with equally likely outcomes, then

\[\mathbb{P}(A) = \frac{|A|}{|S|}.\]

A probability is a function…

\(\mathbb{P}(A)\) is a function with

Input: event \(A\) from the sample space \(S\), (\(A \subseteq S\))
Output: a number between 0 and 1 (inclusive)

\[\mathbb{P}(A): S \rightarrow [0,1]\]

A function that follows some specific rules though!

See Probability Axioms on next slide.

Probability Axioms

Axiom 1

For every event \(A\), \(0\leq\mathbb{P}(A)\leq 1\).

Axiom 2

For the sample space \(S\), \(\mathbb{P}(S)=1\).

Axiom 3

If \(A_1, A_2, A_3, \ldots\), is a collection of disjoint events, then \[\mathbb{P}\Big( \bigcup \limits_{i=1}^{\infty}A_i\Big) = \sum_{i=1}^{\infty}\mathbb{P}(A_i).\]

Some probability properties

Using the Axioms, we can prove all other probability properties!

Proposition 1

For any event \(A\), \(\mathbb{P}(A)= 1 - \mathbb{P}(A^C)\)

Proposition 2

\(\mathbb{P}(\emptyset)=0\)

Proposition 3

If \(A \subseteq B\), then \(\mathbb{P}(A) \leq \mathbb{P}(B)\)

Proposition 4

\(\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) - \mathbb{P}(A \cap B)\)

Proposition 5

\(\mathbb{P}(A \cup B \cup C) = \mathbb{P}(A) + \mathbb{P}(B) + \\ \mathbb{P}(C) - \mathbb{P}(A \cap B) - \mathbb{P}(A \cap C) - \\ \mathbb{P}(B \cap C) + \mathbb{P}(A \cap B \cap C)\)

Proposition 1 Proof

Proposition 1

For any event \(A\), \(\mathbb{P}(A)= 1 - \mathbb{P}(A^C)\)

Proposition 2 Proof

Proposition 2

\(\mathbb{P}(\emptyset)=0\)

Proposition 3 Proof

Proposition 3

If \(A \subseteq B\), then \(\mathbb{P}(A) \leq \mathbb{P}(B)\)

Proposition 4 Visual Proof

Proposition 4

\(\mathbb{P}(A \cup B) = \mathbb{P}(A) + \mathbb{P}(B) - \mathbb{P}(A \cap B)\)

Proposition 5 Visual Proof

Proposition 5

\(\mathbb{P}(A \cup B \cup C) = \mathbb{P}(A) + \mathbb{P}(B) + \mathbb{P}(C) - \mathbb{P}(A \cap B) - \mathbb{P}(A \cap C) - \mathbb{P}(B \cap C) + \mathbb{P}(A \cap B \cap C)\)

Partitions

Definition: Partition

A set of events \(\{A_i\}_{i=1}^{n}\) create a partition of \(A\), if

the \(A_i\)’s are disjoint (mutually exclusive) and
\(\bigcup \limits_{i=1}^n A_i = A\)

Example 2

If \(A \subset B\), then \(\{A, B \cap A^C\}\) is a partition of \(B\).
If \(S = \bigcup \limits_{i=1}^n A_i\), and the \(A_i\)’s are disjoint, then the \(A_i\)’s are a partition of the sample space.

Creating partitions is sometimes used to help calculate probabilities, since by Axiom 3 we can add the probabilities of disjoint events.

Venn Diagram Probabilities

Weekly medications

Example 3

If a subject has an

80% chance of taking their medication this week,
70% chance of taking their medication next week, and
10% chance of not taking their medication either week,

then find the probability of them taking their medication exactly one of the two weeks.

Hint: Draw a Venn diagram labelling each of the parts to find the probability.