Probability Rules and Distributions: Study Notes

Probability Rules

Introduction to Probability

Probability forms the basis of inferential statistics. It quantifies the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain). Understanding probability is essential for analyzing random phenomena and making predictions.

• Probability: The measure of the likelihood that a random outcome or event occurs.

• Sample Space (S): The set of all possible outcomes of a probability experiment.

• Event (E): Any collection of outcomes from the sample space.

Example: Rolling a Single Die

• Sample Space: S = {1, 2, 3, 4, 5, 6}

• Event E ("roll an odd number"): E = {1, 3, 5}

• Event F ("roll an even number"): F = {2, 4, 6}

Example Application: If you roll a die, the probability of rolling an odd number is the number of odd outcomes divided by the total number of outcomes.

Compound Experiments

When multiple random processes are combined, the sample space expands to include all possible ordered outcomes.

• Example: Rolling a die and then flipping a coin. Sample space: {(1,H), (1,T), (2,H), (2,T), ..., (6,H), (6,T)}

Applying the Rules of Probability

Probability Notation and Rules

• P(E): Probability that event E occurs.

• Rule 1: $0 \leq P(E) \leq 1$

• Rule 2: The sum of the probabilities of all outcomes in the sample space must equal 1.

Probability Distributions

A probability distribution lists all possible outcomes and their probabilities. For a valid probability distribution:

• Each probability must be between 0 and 1.

• The sum of all probabilities must be exactly 1.

Outcome	Probability
1	0.15
2	0.20
3	0.25
4	0.10
5	0.25
Total	0.95

Note: Since the probabilities add up to 0.95, not 1, this is not a valid probability distribution.

Empirical Probability

Relative Frequency Approach

Empirical probability is estimated by conducting experiments and observing outcomes.

• Formula: $P(E) = \frac{\text{Frequency of E}}{\text{Number of trials}}$

• Example: If a die is rolled 100 times and a four appears 31 times, $P(\text{rolling a four}) = \frac{31}{100} = 0.31$

Classical Probability

Equally Likely Outcomes

When all outcomes are equally likely, classical probability can be used.

• Formula: $P(E) = \frac{\text{Number of outcomes in E}}{\text{Total number of outcomes}}$

• Example: For rolling a die, the probability of rolling a 3, 5, or 8 (assuming sample space is {1,2,3,5,8,11,13,15}) is $P(E) = \frac{3}{8} = 0.375$

Probability Applications

Probability with Tulip Bulbs

• Example: A bag contains 100 tulip bulbs: 40 red, 35 yellow, 25 white.

• Probability of selecting a red bulb: $P(\text{red}) = \frac{40}{100} = 0.4$

• Probability of selecting a yellow bulb: $P(\text{yellow}) = \frac{35}{100} = 0.35$

Constructing a Probability Distribution (Relative Frequency)

Relative frequency distributions are constructed from observed data.

Response	Frequency	Probability
Never	111	0.026
Rarely	249	0.055
Sometimes	345	0.076
Most times	716	0.158
Always	3073	0.684
Total	4494	1.000

Application: The probability that a college student never wears a seatbelt is 0.026 (2.6%), which is less than the 5% threshold, so it is considered unusual.

Summary Table: Probability Concepts

Concept	Definition	Formula
Probability	Likelihood of an event	$0 \leq P(E) \leq 1$
Sample Space	Set of all possible outcomes	List or set notation
Event	Subset of sample space	List or set notation
Empirical Probability	Based on experiment	$P(E) = \frac{\text{Frequency of E}}{\text{Number of trials}}$
Classical Probability	Equally likely outcomes	$P(E) = \frac{\text{Number of outcomes in E}}{\text{Total outcomes}}$

Additional info: These notes cover foundational probability concepts, including sample spaces, events, probability rules, empirical and classical probability, and probability distributions. Examples and tables are provided for practical understanding.