Chapter 4: Probability Concepts
Welcome to Chapter 4, where we'll be exploring the exciting world of probability! This lesson will cover the basics of probability, events, and some essential rules to help you calculate probabilities effectively. Let's get started!
4.1 Probability Basics
Probability is a measure of the likelihood that an event will occur. It's always a number between 0 and 1, inclusive. Here are some key facts:
- Property 1: $0 \le P(E) \le 1$, where $P(E)$ is the probability of event $E$.
- Property 2: If an event cannot occur, its probability is 0 (impossible event).
- Property 3: If an event must occur, its probability is 1 (certain event).
4.2 Events and Sample Space
- Sample Space: The collection of all possible outcomes of an experiment.
- Event: A collection of outcomes from the sample space (a subset of the sample space).
Example: Consider rolling a fair six-sided die. The sample space is {1, 2, 3, 4, 5, 6}. An event could be rolling an even number, which corresponds to the subset {2, 4, 6}.
4.3 Probability for Equally Likely Outcomes (f/N Rule)
If an experiment has $N$ possible outcomes, all equally likely, and an event can occur in $f$ ways, then the probability of the event is given by:
$$P(\text{event}) = \frac{f}{N} = \frac{\text{Number of ways event can occur}}{\text{Total number of possible outcomes}}$$
Example: When two balanced dice are rolled, there are 36 equally likely outcomes. What is the probability that doubles are rolled? There are 6 possible doubles (1-1, 2-2, 3-3, 4-4, 5-5, and 6-6). Therefore, the probability of rolling doubles is $\frac{6}{36} = \frac{1}{6}$.
4.4 Relationships Among Events
- (not E): The event "E does not occur."
- (A & B): The event "both A and B occur" (denoted as $A \cap B$).
- (A or B): The event "either A or B or both occur" (denoted as $A \cup B$).
4.5 Mutually Exclusive Events
Two or more events are mutually exclusive if no two of them have outcomes in common. If $A$ and $B$ are mutually exclusive, then $P(A \cap B) = 0$.
4.6 Some Rules of Probability
- Special Addition Rule: If events A and B are mutually exclusive, then $P(A \text{ or } B) = P(A) + P(B)$. More generally, if events A, B, C,... are mutually exclusive, then $P(A \text{ or } B \text{ or } C \text{ or } ...) = P(A) + P(B) + P(C) + ...$
- Complementation Rule: For any event E, $P(E) = 1 - P(\text{not } E)$.
- General Addition Rule: If A and B are any two events, then $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.
4.7 Conditional Probability
The probability that event B occurs given that event A has already occurred is called conditional probability, denoted as $P(B | A)$.
Conditional Probability Rule: If A and B are any two events with $P(A) > 0$, then
$$P(B | A) = \frac{P(A \text{ and } B)}{P(A)}$$
4.8 Independent Events
Event B is said to be independent of event A if $P(B | A) = P(B)$. In other words, the occurrence of A does not affect the probability of B.
Special Multiplication Rule (for Two Independent Events): If A and B are independent events, then
$$P(A \text{ and } B) = P(A) \cdot P(B)$$
Probability is a fundamental concept with wide-ranging applications. Keep practicing, and you'll become a probability pro in no time!