Probability and Counting Principles: Core Concepts and Applications

Ch. 5 - Probability

Basic Concepts of Probability

Probability is a measure of how likely an event is to occur, expressed as a number between 0 and 1. The set of all possible outcomes is called the sample space. Probabilities can be determined theoretically (before events occur) or empirically (after events occur, based on data).

• Theoretical Probability: Calculated based on possible outcomes, before any experiment is performed.

• Empirical (Experimental) Probability: Calculated after performing an experiment, based on observed outcomes.

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

• Example: When rolling a six-sided die, the probability of rolling a number greater than 3 is $P(>3) = \frac{3}{6} = 0.5$.

Complements

The complement of an event A (denoted as A', Ac, or not A) consists of all outcomes in the sample space that are not in A. The sum of the probabilities of an event and its complement is always 1.

• Formula: $P(A') = 1 - P(A)$

• Example: The probability of not rolling a 4 on a six-sided die is $P(\text{not 4}) = 1 - P(4) = 1 - \frac{1}{6} = \frac{5}{6}$.

Addition Rule

The addition rule is used to find the probability that at least one of two events occurs. The rule differs for mutually exclusive and non-mutually exclusive events.

• Mutually Exclusive Events: Events that cannot occur at the same time. $P(A \cap B) = 0$.

• Formula (Mutually Exclusive): $P(A \cup B) = P(A) + P(B)$

• Non-Mutually Exclusive Events: Events that can occur together. $P(A \cap B) \neq 0$.

• Formula (General): $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• Example: Drawing a card that is a diamond or a king from a standard deck: $P(\text{diamond or king}) = P(\text{diamond}) + P(\text{king}) - P(\text{diamond and king})$

Multiplication Rule: Independent Events

For independent events (where the outcome of one does not affect the other), the probability that both events occur is the product of their probabilities.

• Formula: $P(A \cap B) = P(A) \times P(B)$

• Example: Probability of getting heads on two consecutive coin flips: $P(H \text{ and } H) = P(H) \times P(H) = 0.5 \times 0.5 = 0.25$.

Multiplication Rule: Dependent Events

For dependent events (where the outcome of one affects the other), the probability that both occur is the probability of the first event times the conditional probability of the second event given the first.

• Formula: $P(A \cap B) = P(A) \times P(B|A)$

• Conditional Probability: $P(B|A) = \frac{P(A \cap B)}{P(A)}$

• Example: Drawing two aces from a deck without replacement: $P(\text{first ace}) = \frac{4}{52}$, $P(\text{second ace}|\text{first ace}) = \frac{3}{51}$, so $P(\text{both aces}) = \frac{4}{52} \times \frac{3}{51}$.

Contingency Tables

Contingency tables display the frequency distribution of variables and are used to calculate marginal, joint, and conditional probabilities.

• Marginal Probability: Probability of a single event occurring.

• Joint Probability: Probability of two events occurring together.

• Conditional Probability: Probability of one event given another has occurred.

Drives a Car	Yes	No	Total
Senior	40	10	50
Junior	20	30	50
Total	60	40	100

Bayes' Theorem

Bayes' Theorem allows us to find the probability of an event given new information, especially when direct calculation is difficult.

• Formula: $P(B|A) = \frac{P(A|B) \times P(B)}{P(A|B) \times P(B) + P(A|B') \times P(B')}$

• Example: If a test for a disease is 95% accurate and the disease is rare, Bayes' Theorem helps determine the probability a person actually has the disease given a positive test result.

Counting Principles

Fundamental Counting Principle

The Fundamental Counting Principle states that if one event can occur in m ways and a second in n ways, the two events together can occur in $m \times n$ ways. This extends to more than two events by multiplying the number of options for each event.

• Example: If you have 3 shirts and 4 pants, the number of outfits is $3 \times 4 = 12$.

Permutations

Permutations are arrangements of objects where order matters. The number of permutations of n objects taken r at a time is:

• Formula: $P(n, r) = \frac{n!}{(n-r)!}$

• Example: The number of ways to arrange 5 shirts over 5 days: $5! = 120$.

Permutations of Non-Distinct Objects

When some objects are identical, the number of unique permutations is reduced. The formula is:

• Formula: $\frac{n!}{r_1! r_2! \ldots r_k!}$, where $r_1, r_2, \ldots$ are counts of each identical object.

• Example: Arranging the letters in "BANANA": $\frac{6!}{3!2!1!}$

Combinations

Combinations are selections of objects where order does not matter. The number of combinations of n objects taken r at a time is:

• Formula: $C(n, r) = \frac{n!}{r!(n-r)!}$

• Example: The number of ways to choose 2 flavors from 32: $C(32, 2)$

Permutations vs. Combinations

Type	Order Matters?	Formula
Permutation	Yes	$P(n, r) = \frac{n!}{(n-r)!}$
Combination	No	$C(n, r) = \frac{n!}{r!(n-r)!}$

Applications in Probability

Counting principles are often used to determine the number of possible outcomes in probability problems, such as lottery odds, arrangements, and selections.

• Example: The probability of winning a lottery by matching 5 numbers out of 40 is $P(\text{win}) = \frac{1}{C(40, 5)}$