Fundamentals of Probability and Probability Distributions

Probability: Basic Concepts and Definitions

Key Terms in Probability

Understanding the foundational vocabulary of probability is essential for analyzing random experiments and interpreting results.

• Probability: The measure of how likely an event is to occur, expressed as a number between 0 and 1. A probability of 0 means the event cannot occur, while a probability of 1 means the event is certain to occur.

• Experiment: A process or action that results in one or more outcomes. For example, tossing a coin or rolling a die.

• Sample Space: The set of all possible outcomes of an experiment. Denoted as S. Example: For tossing two coins, S = {HH, HT, TH, TT}.

• Outcome: A single possible result of an experiment. Example: Getting 'HT' when tossing two coins.

• Event: Any subset of the sample space. An event may consist of one or more outcomes. Example: Getting at least one tail when tossing two coins: {HT, TH, TT}.

• Equally Likely: Outcomes that have the same probability of occurring. Example: When tossing a fair coin, 'Head' and 'Tail' are equally likely.

• Complement: The set of all outcomes in the sample space that are not in the event. If event A occurs, its complement A' does not occur. Example: If A is 'getting at least one tail', then A' is 'getting no tails' (i.e., {HH}).

Calculating Probabilities in Simple Experiments

Probability of Events in Coin Tosses

When analyzing experiments like tossing coins, we use the sample space to determine the probability of various events.

• Sample Space for Two Coins: S = {HH, HT, TH, TT}

• Event A: Getting at most one tail (i.e., zero or one tail). A = {HH, HT, TH} $P(A) = \frac{3}{4}$

• Event B: Getting all tails. B = {TT} $P(B) = \frac{1}{4}$

• Event C: Getting more than one tail. C = {TT} $P(C) = \frac{1}{4}$

• Event D: Getting a head on the first roll. D = {HH, HT} $P(D) = \frac{2}{4} = \frac{1}{2}$

• Event E: Getting at least one tail in two flips. E = {HT, TH, TT} $P(E) = \frac{3}{4}$

Note: For equally likely outcomes, $P(A) = \frac{\text{Number of outcomes in A}}{\text{Total number of outcomes in S}}$

Probability with Numbers: Sample Space of Integers

Consider the sample space S as the set of whole numbers starting at 0 and less than 20: S = {0, 1, 2, ..., 19}.

• Event A: Even numbers. A = {0, 2, 4, 6, 8, 10, 12, 14, 16, 18} $P(A) = \frac{10}{19}$

• Event B: Numbers greater than 13. B = {14, 15, 16, 17, 18, 19} $P(B) = \frac{6}{19}$

• Event A AND B: Even numbers greater than 13. A ∩ B = {14, 16, 18} $P(A \cap B) = \frac{3}{19}$

• Event A OR B: Numbers that are even or greater than 13. A ∪ B = {0, 2, 4, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19} $P(A \cup B) = \frac{13}{19}$

• Event C: The outcome is 25. C = {25} (not in S) $P(C) = 0$

Probability Distributions

Discrete Probability Distributions

A probability distribution assigns probabilities to each possible value of a discrete random variable. The sum of all probabilities must be 1.

• Example Table: Let X = number of days attending practice.

X	% of days	P(X)
0	2%	0.02
1	8%	0.08
2	90%	0.90

• Properties:

  • Each $P(X)$ is between 0 and 1.

  • The sum of all $P(X)$ values is 1: $\sum P(X) = 1$

Probability Distribution Table Example

X	P(X)
1	0.12
2	0.18
3	0.30
4	0.25
5	0.10
6	0.05

• Sum of Probabilities: $0.12 + 0.18 + 0.30 + 0.25 + 0.10 + 0.05 = 1$

• Probability X ≥ 5: $P(X \geq 5) = P(5) + P(6) = 0.10 + 0.05 = 0.15$

• Expected Value (Mean): The expected value of X is $E(X) = \sum X \cdot P(X)$ $E(X) = 1 \times 0.12 + 2 \times 0.18 + 3 \times 0.30 + 4 \times 0.25 + 5 \times 0.10 + 6 \times 0.05 = 3.23$ years

Additional Probability Rules and Concepts

Mutually Exclusive Events

Two events are mutually exclusive if they cannot both occur at the same time. That is, $A \cap B = \emptyset$ and $P(A \cap B) = 0$.

• Example: Getting all heads and getting all tails in two coin tosses are mutually exclusive events.

Addition Rule for Probability

For any two events A and B:

• If A and B are mutually exclusive: $P(A \cup B) = P(A) + P(B)$

• If A and B are not mutually exclusive: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

Complement Rule

The probability that event A does not occur is $P(A') = 1 - P(A)$.

Expected Value (Mean) of a Discrete Random Variable

The expected value (mean) of a discrete random variable X is:

• $E(X) = \sum X \cdot P(X)$

• Example: If X can take values 1, 2, 3 with probabilities 0.2, 0.5, 0.3, then $E(X) = 1 \times 0.2 + 2 \times 0.5 + 3 \times 0.3 = 2.1$

Summary Table: Key Probability Terms

Term	Definition	Example
Probability	Measure of likelihood of an event (0 ≤ P ≤ 1)	P(getting heads in a coin toss) = 0.5
Experiment	Process with uncertain outcome	Rolling a die
Sample Space	Set of all possible outcomes	{1,2,3,4,5,6} for a die
Outcome	Single result of an experiment	Rolling a 4
Event	Subset of sample space	Getting an even number: {2,4,6}
Equally Likely	All outcomes have same probability	Each side of a fair die
Complement	All outcomes not in the event	Not rolling a 6: {1,2,3,4,5}

Additional info: Some context and examples were inferred and expanded for clarity and completeness, especially in the definitions and probability distribution sections.