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# Introduction to Probability ## §1. Sample Space **Definition:** The set of all possible outcomes of a random experiment is called the sample space, denoted by S. Each outcome in the sample space is called a sample point. **Example:** 1. Toss a coin, $S = \{H, T\}$ 2. Toss a die, $S = \{1, 2...

# Introduction to Probability ## §1. Sample Space **Definition:** The set of all possible outcomes of a random experiment is called the sample space, denoted by S. Each outcome in the sample space is called a sample point. **Example:** 1. Toss a coin, $S = \{H, T\}$ 2. Toss a die, $S = \{1, 2, 3, 4, 5, 6\}$ 3. Toss a coin twice, $S = \{(H, H), (H, T), (T, H), (T, T)\}$ ## §2. Event **Definition:** An event is a subset of the sample space. An event is said to occur if the outcome of the experiment is a sample point in the event. **Example:** Toss a die, $S = \{1, 2, 3, 4, 5, 6\}$ Event A: observe an even number, $A = \{2, 4, 6\}$ Event B: observe a number greater than 4, $B = \{5, 6\}$ ## §3. Probability ### 3.1 Definition Probability is a way of assigning a number between 0 and 1 to each event, representing the likelihood of the event occurring. ### 3.2 Axioms of Probability 1. For any event A, $P(A) \ge 0$ 2. $P(S) = 1$ 3. For any sequence of mutually exclusive events $A_1, A_2,...$, $P(A_1 \cup A_2 \cup...) = \sum_{i=1}^{\infty} P(A_i)$ ### 3.3 Some Basic Probability Laws 1. $P(\phi) = 0$ 2. $P(A^c) = 1 - P(A)$ 3. $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ 4. If $A \subset B$, then $P(A) \le P(B)$