Unit 1: Probability models and axioms

1. Lec. 1: Probability models and axioms

1.1. Sample Space

• Definition: the set of all of possible outcomes is called the sample space of the experiment,. Denoted by $\Omega$
• A subset of the sample space is called event. Probability is assigned to events.
• sample space can be finite, infinite, continuous, etc.
• The construction of a sample space is a description of the possible outcomes of a probabilistic experiment.

Examples

• discrete/finite example:
• Two rolls of a tetrahedral die.
• roll twice, get 16 outcomes. (the order sometimes matters)
• continuous example:
• (x, y) such that $0 \le x, y \le 1$: a square box.

1.2. Probability axioms

• Non-negativity: $P(A) \ge 0$
• Normalization: $P(\Omega) = 1$
• (Finite) additivity: $\text{If } A \cap B = \emptyset,\text{then } P(A \cup B) = P(A) + P(B)$
• $\emptyset$ denote empty set.
• $A \cap B = \emptyset$ means A and B are disjoint events.

Consequences

• $P(A) \le 1$
• $P(\emptyset) = 0$
• $P(A) + P(A^c) = 1$
• $P(A \cup B \cup C) = P(A) + P(B) + P(C)$
• A, B and C are disjoint events.
• and similarly for k disjoint events.
• $P({s_1, s_2, \ldots, s_k}) = P({s_1} \cup {s_2} \cup \ldots \cup {s_k})= P({s_1}) + P({s_2}) + \ldots + P({s_k})$
• $\text{If } A \subset B = \phi,\text{then } P(A) \le P(B)$
• $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• $P(A \cup B) \le P(A) + P(B)$
• $P(A \cup B \cup C) = P(A) + P(A^c \cap B) + P(A^c \cap B^c \cap C)$

1.3. Discrete Models

• Example: Coin tosses. Consider an experiment involving a single coin toss. There are two possible outcomes, heads (H) and tails (T). The sample space is $\Omega = \{H, T\}$, and the events are $\{H, T\}, \{H\}, \{T\}, \emptyset.$

Discrete Probability Law

• the probability of any event $\{s_1, s_2, \ldots, s_n \}$ is the sum of the probabilities of its elements: $P(\{s_1, s_2, \ldots, s_n \}) = P(\{s_1\}) + P(\{s_2\}) + \ldots + P(\{s_n\})$

Discrete Uniform Law

• Assume $\Omega$ consists of n equals likely elements
• Then the probability of any event A is given by $P(A) = \frac{\text{Number of elements of} A}{n}$

1.4. Continuous Models

• Example: (x, y) such that $0 \le x, y \le 1$
• $P({(x, y)| x + y \le 1/2}) = \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8}$
• Because it's a right triangle with length 1/2 of each side.
• $P({(0.5, 0.3)}) = 0$
• Because it's the area of a single point.

Probability calculation: discrete but infinite sample space

• Keep tossing a coin and the outcome is the number of tosses until we observe heads for the first time.
• Probability: $P(n) = \frac{1}{2^n},\ n = 1,2,\ldots$
• Legitimate check: $P(\Omega) = 1$
• $\sum_{n = 1}^{\infty} \frac{1}{2^n} = \frac{1}{2}\sum_{n = 0}^{\infty} \frac{1}{2^n} = \frac{1}{2} \cdot \frac{1}{1-1/2} = 1$
• Geometric Series Theorem
• $P(\text{outcome is even}) = P(\{2, 4, 6, \ldots\}) = P(\{2\}), P(\{4\}), P(\{6\}), \ldots\}) = \frac{1}{2^2} + \frac{1}{2^4} + \frac{1}{2^6} + \cdots = \frac{1}{4} \cdot \frac{1}{1 - \frac{1}{4} } = \frac{1}{3}$
• To support this calculation, we'll need:

• If $A_1, A_2, \ldots$ is an infinite sequence of disjoint events, then $P(A_1 \cup A_2 \cup \cdots) = P(A_1) + P(A_2) + \cdots$
• Notice the events should be sequence, like the example above.
• Example (x, y) such that $0 \le x, y \le 1$ can not be arranged in a sequence. and the elements are NOT countable.
• Additivity holds only for "countable" sequences of events.

Countable and uncountable sets

• Countable: can be put in 1-1 correspondence with positive integers
• positive integers: $1, 2, 3, \ldots$
• integers: $0, 1, -1, 2, -2, \ldots$
• pairs of positive integers
• rational numbers q with 0 < q < 1
• $1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, \ldots$
• Uncountable:
• the interval [0, 1]
• the reals, the plane...

1.6. Interpretations of probability theory

• A framework for analyzing phenomena with uncertain outcomes
• Rules for consistent reasoning
• Used for predictions and decisions

2. Words

• tetrahedral [,tetrə'hi:drəl, -'he-] adj. 四面体的；有四面的
• axiom ['æksiəm] n. [数] 公理；格言；自明之理