Probability and Counting Rules – Study Notes (Triola, Elementary Statistics Ch. 4)

Basic Concepts of Probability

Introduction to Probability

Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means the event cannot happen and 1 means the event is certain to happen. The set of all possible outcomes is called the sample space.

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

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

• Probability Formula: $P(E) = \frac{\text{Number of ways event E can occur}}{\text{Total number of outcomes in the sample space}}$

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

Practice Problems

• Given a group, probability of selecting a person wearing jeans: $P(\text{jeans}) = \frac{\text{number wearing jeans}}{\text{total in group}}$

• Probability of picking a quarter from a purse with 3 quarters, 4 nickels, and 2 dimes: $P(\text{quarter}) = \frac{3}{9}$

Complements

Complementary Events

The complement of an event A (written as A', Ac, or not A) consists of all outcomes where A does not occur. The sum of the probabilities of an event and its complement is always 1:

• $P(A) + P(A') = 1$

• $P(A') = 1 - P(A)$

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

Practice Problems

• Probability of not drawing a queen from a deck: $P(\text{not queen}) = 1 - \frac{4}{52} = \frac{48}{52}$

• If a red or yellow marble is drawn 6 out of 8 times, probability of green: $P(\text{green}) = 1 - \frac{6}{8} = \frac{2}{8}$

Addition Rule

Probability of Mutually Exclusive Events

Events are mutually exclusive if they cannot happen at the same time. For such events, the probability that either event A or event B occurs is:

• $P(A \cup B) = P(A) + P(B)$

• Example: Probability of rolling a 3 or a 5 on a die: $P(3 \text{ or } 5) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6}$

Probability of Non-Mutually Exclusive Events

If events are not mutually exclusive (they can occur together), subtract the probability of their intersection:

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

• Example: Probability of rolling a number greater than 3 or an even number on a die.

Practice Problems

• Probability of drawing a diamond or a king from a deck: $P(\text{diamond}) + P(\text{king}) - P(\text{diamond and king})$

Multiplication Rule

Independent Events

Events are independent if the occurrence of one does not affect the other. For independent events A and B:

• $P(A \cap B) = P(A) \times P(B)$

• Example: Probability of getting heads on two coin flips: $P(H \text{ and } H) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Dependent Events

Events are dependent if the occurrence of one affects the probability of the other. For dependent events:

• $P(A \cap B) = P(A) \times P(B|A)$

• Example: Drawing two marbles without replacement from a bag.

Practice Problems

• Probability of drawing two nonfiction books from a list of 100 (62 fiction, 38 nonfiction): $P(\text{first nonfiction}) \times P(\text{second nonfiction | first nonfiction})$

Conditional Probability

Definition and Formula

Conditional probability is the probability of event B occurring given that event A has occurred:

• $P(B|A) = \frac{P(A \cap B)}{P(A)}$

• Example: Probability a student has a math major given they have a science major.

Bayes' Theorem

Bayes' Theorem

Bayes' Theorem allows us to find the probability of an event based on prior knowledge of conditions related to the event:

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

• Example: Probability a marble came from the left bag given it is red, using prior probabilities and conditional probabilities.

Practice Problems

• Medical test accuracy, train delays with precipitation, etc.

Fundamental Counting Principle

Counting Outcomes

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. For more events, multiply the number of options for each event.

• Example: 3 shirts and 4 pants: $3 \times 4 = 12$ outfits.

• Example: 4 appetizers and 6 entrees: $4 \times 6 = 24$ meal combinations.

Permutations and Combinations

Permutations

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

• $P(n, r) = \frac{n!}{(n-r)!}$

• Example: Arranging 5 shirts over 5 days: $5! = 120$ ways.

Permutations of Non-Distinct Objects

When some objects are identical, divide by the factorial of the number of identical objects:

• $\text{Number of arrangements} = \frac{n!}{r_1! r_2! \ldots r_k!}$

• 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:

• $C(n, r) = \frac{n!}{r!(n-r)!}$

• Example: Choosing 2 flavors from 32: $C(32, 2)$

Permutations vs. Combinations

• Permutations: Order matters (e.g., arranging people in a line).

• Combinations: Order does not matter (e.g., selecting a team).

Contingency Tables

Finding Probabilities from Contingency Tables

A contingency table displays frequencies for combinations of two or more categorical variables. Probabilities can be found as follows:

• Marginal Probability: Probability of a single event (row or column total divided by grand total).

• Joint Probability: Probability of two events occurring together (cell frequency divided by grand total).

• Conditional Probability: Probability of one event given another (cell frequency divided by row or column total).

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

Example: Probability a randomly selected student is a senior and drives a car: $P(\text{Senior} \cap \text{Drives a Car}) = \frac{40}{100}$

Summary Table: Probability Rules

Rule	Formula	When to Use
Complement	$P(A') = 1 - P(A)$	Finding probability of 'not A'
Addition (Mutually Exclusive)	$P(A \cup B) = P(A) + P(B)$	Events cannot occur together
Addition (Not Mutually Exclusive)	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$	Events can occur together
Multiplication (Independent)	$P(A \cap B) = P(A) \times P(B)$	Events do not affect each other
Multiplication (Dependent)	$P(A \cap B) = P(A) \times P(B|A)$	Events affect each other
Conditional Probability	$P(B|A) = \frac{P(A \cap B)}{P(A)}$	Probability of B given A
Bayes' Theorem	$P(B|A) = \frac{P(A|B) \times P(B)}{P(A|B) \times P(B) + P(A|B') \times P(B')}$	Reverse conditional probability