Welcome to Chapter 6: Probability, Randomness, and Uncertainty!

Get ready to explore the fascinating world of probability! This chapter will cover sections 6.1 and 6.2, focusing on the introduction to probability and the addition rules for probability. We'll break down key concepts with examples to make sure you grasp the material.

6.1 Introduction to Probability

Let's start with some basic definitions:

  • Random Experiment: An activity or phenomenon with uncertain outcomes, where all possible outcomes can be specified. Think of tossing a coin!
  • Sample Space (S): The set of all possible distinct outcomes of a random experiment. For example, when tossing a coin, $S = \{Head, Tail\}$.
  • Outcome: Any member of the sample space.
  • Event: A set of outcomes.

Consider a few examples:

  • Experiment 1: Tossing a coin once. The sample space is $S = \{Head, Tail\}$.
  • Experiment 2: Inspecting a transistor. The sample space is $S = \{Pass, Fail\}$.
  • Experiment 3: Drawing a card from a standard deck. The sample space consists of the four suits: Hearts, Clubs, Spades, and Diamonds.

There are two approaches to probability:

  • Relative Frequency (Experimental) Approach: If an experiment is performed $n$ times under identical conditions, and event $A$ happens $k$ times, the relative frequency of $A$ is given by: $$Relative Frequency of A = \frac{k}{n}$$ If the relative frequency converges as $n$ increases, it is considered the probability of $A$.
  • Classical Probability (Theoretical) Approach: The probability of an event $A$, denoted as $P(A)$, is given by: $$P(A) = \frac{\text{number of outcomes in A}}{\text{total number of outcomes in the sample space}}$$.

Important Probability Laws:

  1. A probability of zero means the event cannot happen.
  2. A probability of one means the event must happen.
  3. All probabilities must be between zero and one, inclusively: $0 \le P(A) \le 1$.
  4. The sum of the probabilities of all outcomes must equal one.

6.2 Addition Rules for Probability

Now, let's move on to compound events and addition rules.

  • Compound Event: An event defined by combining two or more events.
  • Union (A or B): The set of outcomes that are included in event A, event B, or both. Denoted as $A \cup B$.
  • Intersection (A and B): The set of outcomes that are included in both event A and event B. Denoted as $A \cap B$.
  • Complement (A'): The set of all outcomes in the sample space that are not in A. The probability is given by $P(A') = 1 - P(A)$.

Key Definitions:

  • Mutually Exclusive Events: Two events are mutually exclusive if they have no outcomes in common. If A and B are mutually exclusive, then $P(A \cap B) = 0$.

Probability Laws:

  • Union of Mutually Exclusive Events: If events A and B are mutually exclusive, then $P(A \cup B) = P(A) + P(B)$.
  • General Addition Rule: For any two events A and B, $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$.

Keep practicing and you'll master these concepts in no time! Good luck with your studies!