Chapter 3: Probability – Basic Concepts and Counting in Statistics

Probability: Basic Concepts and Counting

Definitions and Basic Set Theory

Probability is a fundamental concept in statistics that quantifies the likelihood of events occurring in random experiments. Understanding the language and structure of probability is essential for analyzing random phenomena.

• Probability: A number between 0 and 1 representing how likely it is that a given event will occur.

• Random Experiment: An experiment whose outcome is not known until it is observed.

• Sample Space (S): The set of all possible outcomes of a random experiment. Each outcome is listed only once.

• Sample Point: An individual element of the sample space.

• Event: Any subset of the sample space; a collection of outcomes.

Example: Suppose you roll a die. The sample space is S = {1, 2, 3, 4, 5, 6}. A sample point could be 3. If P is the event that the number on the top of the die is prime, then P = {2, 3, 5}.

Counting Techniques

Counting techniques are used to determine the number of possible outcomes in a sample space, which is essential for calculating probabilities.

• Fundamental Counting Principle (Multiplication Rule for Counting): If there are k sets with n1, n2, ..., nk elements respectively, the total number of ways to choose one element from each set is n1 × n2 × ... × nk.

Examples:

• Sample space for a pair of dice: 6 × 6 = 36 outcomes.

• Dining selections: 3 salads × 5 main courses × 2 desserts = 30 possible meals.

• Arranging 5 books: 5! = 120 ways.

• 5-digit codes (no repetition, first digit not zero): 9 × 9 × 8 × 7 × 6 = 27,216 possible codes.

Probability

Probability quantifies the chance of an event occurring and can be defined in several ways:

• Classical (Theoretical) Probability: If all outcomes are equally likely, the probability of event A is:

$P(A) = \frac{n(A)}{n(S)}$

• Empirical (Experimental) Probability: Based on observed data, the probability of event B is:

$P(B) = \frac{f}{n}$

• Law of Large Numbers: As an experiment is repeated, the empirical probability approaches the theoretical probability.

• Probability values range from 0 (impossible event) to 1 (certain event).

• The probability of the sample space is 1: $P(S) = 1$.

• Mutually exclusive events: $A \cap B = \emptyset$.

• If $E_1, E_2, ..., E_n$ are mutually exclusive and exhaustive, $P(E_1 \cup E_2 \cup ... \cup E_n) = P(E_1) + P(E_2) + ... + P(E_n) = 1$.

The “TESS” Approach

The TESS approach is a systematic method for solving probability problems:

• T: Translate the problem from words to statistical form.

• E: Expand the statistical form using formulas.

• S: Substitute the values into the expanded form.

• S: Simplify and write the probability as a decimal (rounded to three decimal places).

Example: Let S = {1,2,3,4,5,6,7,8,9,10}. If E = {1,2,3}, then $P(E) = \frac{3}{10} = 0.300$.

Decks of Cards

Understanding the structure of a standard deck of cards is crucial for solving probability problems involving cards.

• One deck has 52 cards (no Jokers).

• Two colors: 26 red (hearts, diamonds), 26 black (spades, clubs).

• Three categories: Ace cards, Number cards, Face (Picture) cards.

• Four suits: Diamonds (♦), Spades (♠), Hearts (♥), Clubs (♣).

• Each suit has 13 cards: A, 2–10, J, Q, K.

• Face cards: J, Q, K (12 in total).

• Prime number cards: 2, 3, 5, 7 (4 per suit, 16 total).

• Even number cards: 2, 4, 6, 8, 10 (20 total).

• Odd number cards: 3, 5, 7, 9 (16 total).

Probability of Drawing Cards

When drawing a card at random from a standard deck, probabilities can be calculated using the classical definition.

• Probability of drawing a six: $P(6) = \frac{4}{52} = 0.077$

• Probability of drawing a face card: $P(\text{face card}) = \frac{12}{52} = 0.231$

• Probability of drawing a diamond: $P(\text{diamond}) = \frac{13}{52} = 0.250$

• Probability of drawing a red queen: $P(\text{red queen}) = \frac{2}{52} = 0.038$

Probability of a Complement

The complement of an event A, denoted Ac or A', is the set of outcomes not in A. The probability of the complement is:

$P(A') = 1 - P(A)$

• $P(A) + P(A') = 1$

• $A \cup A^c = S$ and $P(A \cup A^c) = 1$

• $A \cap A^c = \emptyset$ and $P(A \cap A^c) = 0$

Example: If the probability of obtaining a sum of 11 when rolling two dice is $\frac{1}{18}$, then the probability of not obtaining a sum of 11 is $1 - \frac{1}{18} = \frac{17}{18}$.

Odds of an Event

The odds of an event E is the ratio of the probability that E will occur to the probability that E will not occur.

• Odds in favor of E: $\text{Odds}(E) = n(E) : n(E^c)$

• Odds against E: $\text{Odds}(E^c) = n(E^c) : n(E)$

• Odds can be interpreted as "successes : failures".

Example: If $n(E) = 3$ and $n(S) = 7$, then $n(E^c) = 4$, $P(E) = \frac{3}{7}$, $P(E^c) = \frac{4}{7}$, Odds(E) = 3:4, Odds(Ec) = 4:3.

Tree Diagrams

Tree diagrams are visual tools used to represent all possible outcomes of a sequence of events. Each branch represents a possible outcome, and probabilities can be assigned to each branch.

• Each unique outcome is represented by a path from the start to the end of the tree.

• The total number of branches equals the total number of outcomes.

• Probabilities along branches can be multiplied to find the probability of a sequence of events.

Example: If a jar contains 3 red and 2 blue marbles, and two marbles are drawn without replacement, a tree diagram can show all possible sequences and their probabilities.

Summary Table: Structure of a Standard Deck of Cards

Suit	Color	Number of Cards	Face Cards
Hearts	Red	13	J, Q, K
Diamonds	Red	13	J, Q, K
Clubs	Black	13	J, Q, K
Spades	Black	13	J, Q, K

Additional info: The above table summarizes the main properties of a standard deck, which is essential for probability calculations involving cards.