Probability and Counting Principles – Study Notes

Probability: Basic Concepts

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.

• Probability of an Event (P(E)): The ratio of the number of favorable outcomes to the total number of possible outcomes.

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

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

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

Theoretical vs. Empirical Probability

• Theoretical Probability: Based on what could happen, calculated before events occur using known possible outcomes.

• Empirical (Experimental) Probability: Based on what did happen, calculated after events occur using observed data.

• Example: If a die is rolled 10 times and a number greater than 3 appears 6 times, the empirical probability is $P(>3) = \frac{6}{10} = 0.6$.

Practice Problems

• Given a group, the probability of a randomly selected person wearing jeans is $P(\text{jeans}) = \frac{\text{number wearing jeans}}{\text{total in group}}$.

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

Complementary Events

Definition and Properties

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

• Formula: $P(A) + P(A') = 1$

• Complement Probability: $P(A') = 1 - P(A)$

• Example: Probability of not rolling a 4 on a 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 for Probability

Mutually Exclusive Events

Events are mutually exclusive if they cannot happen at the same time. The probability of either event A or event B occurring is the sum of their probabilities.

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

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

Non-Mutually Exclusive Events

Events are not mutually exclusive if they can occur at the same time. In this case, subtract the probability of both events occurring together to avoid double-counting.

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

• 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: $P(\text{diamond or king}) = P(\text{diamond}) + P(\text{king}) - P(\text{diamond and king})$

Multiplication Rule for Probability

Independent Events

Events are independent if the occurrence of one does not affect the probability of the other. The probability of both events A and B occurring 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: $P(H \text{ and } H) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Dependent Events

Events are dependent if the occurrence of one affects the probability of the other. Use conditional probability for the second event.

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

• Example: Drawing a blue marble, keeping it, then drawing a red marble from a bag of 2 red and 4 blue marbles.

Conditional Probability

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

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

Contingency Tables

Finding Probabilities

A contingency table displays frequencies for combinations of two or more categorical variables. Probabilities can be found as marginal (total), joint (intersection), or conditional (given one event).

• 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

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

Bayes' Theorem

Bayes' Theorem for Conditional Probability

Bayes' Theorem allows us to find the probability of an event given new evidence, even if we do not know all conditional probabilities directly.

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

• Example: Given test accuracy and disease prevalence, find the probability a person has a disease given a positive test result.

Counting Principles

Fundamental Counting Principle

The Fundamental Counting Principle states that if there are m ways to do one thing and n ways to do another, there are $m \times n$ ways to do both.

• Formula: $\text{Total outcomes} = m \times n$

• Example: If you have 3 shirts and 4 pants, you have $3 \times 4 = 12$ possible outfits.

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: Number of ways to arrange 5 shirts over 5 days: $5! = 120$

Permutations of Non-Distinct Objects

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

• Formula: $\frac{n!}{r_1! r_2! \ldots r_k!}$

• Example: Arrangements of the word 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: Number of ways to choose 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).

Summary Table: Probability Rules

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