Welcome to Sections 4.1 - 4.3!
In these sections, we will explore the fundamental concepts of probability. Get ready to dive into sample spaces, events, and the rules that govern probabilities. Let's make learning probability fun and engaging!
Definition 4.1: Probability for Equally Likely Outcomes ($f/N$ Rule)
Suppose an experiment has $N$ possible outcomes, all equally likely. If an event can occur in $f$ ways, then the probability of that event occurring is given by:
$$P( ext{event}) = \frac{f}{N} = \frac{\text{Number of ways event can occur}}{\text{Total number of possible outcomes}}$$Example: When two balanced dice are rolled, there are 36 equally likely outcomes. The probability that the sum of the dice is 11 can be calculated as follows: there are two combinations that result in 11 (5+6 and 6+5), therefore the probability is $2/36 \approx 0.055$, or 5.5%. The probability that doubles are rolled is calculated as follows: there are 6 combinations that result in doubles (1+1, 2+2, 3+3, 4+4, 5+5, 6+6), therefore the probability is $6/36 \approx 0.167$, or 16.7%.
Definition 4.2: Sample Space and Event
- Sample Space: The collection of all possible outcomes for an experiment.
- Event: A collection of outcomes for the experiment; any subset of the sample space. An event occurs if the outcome of the experiment is a member of the event.
Example: Consider the experiment of randomly selecting one card from a deck of 52 playing cards. The sample space is the set of all 52 cards. An event could be selecting a king of hearts.
Definition 4.3: Relationships Among Events
- (not E): The event "E does not occur."
- (A & B): The event "both A and B occur."
- (A or B): The event "either A or B or both occur."
Definition 4.4: Mutually Exclusive Events
Two or more events are mutually exclusive if no two of them have outcomes in common. For example, you cannot draw a card that is both a heart and a spade at the same time, but you can draw a card that is both a heart and a face card. Drawing a Heart and Drawing a Spade are mutually exclusive events.
Key Fact 4.1: Basic Properties of Probabilities
- The probability of an event is always between 0 and 1, inclusive. That is, $0 \le P(E) \le 1$.
- The probability of an event that cannot occur is 0. (An event that cannot occur is called an impossible event.)
- The probability of an event that must occur is 1. (An event that must occur is called a certain event.)
Definition 4.5: Probability Notation
If $E$ is an event, then $P(E)$ represents the probability that event $E$ occurs. It is read "the probability of $E$."
Formula 4.1: The Special Addition Rule
If event $A$ and event $B$ are mutually exclusive, then:
$$P(A \text{ or } B) = P(A) + P(B)$$More generally, if events $A$, $B$, $C$, ... are mutually exclusive, then:
$$P(A \text{ or } B \text{ or } C \text{ or } ...) = P(A) + P(B) + P(C) + ...$$Formula 4.3: The General Addition Rule
If $A$ and $B$ are any two events, then:
$$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$Formula 4.2: The Complementation Rule
For any event $E$,
$$P(E) = 1 - P(\text{not } E)$$Remember, practice makes perfect! Work through examples and exercises to solidify your understanding of these concepts. Keep up the great work!