Probability and Counting Principles – Study Notes

Probability and Counting Principles

Basic Concepts of 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 performing an experiment, based on observed outcomes.

Formula:

$P(E) = \frac{\text{Number of ways event E can occur}}{\text{Total number of possible outcomes}}$

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 (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.

• 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.

• Mutually Exclusive Events: Events that cannot happen at the same time.

• Non-Mutually Exclusive Events: Events that can occur together.

Formulas:

• For mutually exclusive events A and B:

$P(A \cup B) = P(A) + P(B)$

• For non-mutually exclusive events A and B:

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

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

$P(>3 \cup \text{even}) = P(>3) + P(\text{even}) - P(>3 \cap \text{even})$

Multiplication Rule: 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:

$P(H \cap H) = P(H) \times P(H) = 0.5 \times 0.5 = 0.25$

Multiplication Rule: Dependent Events

Events are dependent if the occurrence of one affects the probability of the other. For dependent events, multiply the probability of the first event by the conditional probability of the second event given the first has occurred.

Formula:

$P(A \cap B) = P(A) \times P(B|A)$

Example: Drawing two aces from a deck without replacement:

$P(\text{first ace}) = \frac{4}{52}$ $P(\text{second ace} | \text{first ace}) = \frac{3}{51}$ $P(\text{both aces}) = \frac{4}{52} \times \frac{3}{51}$

Conditional Probability

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

Formula:

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

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

$P(\text{Math} | \text{Science}) = \frac{P(\text{Math} \cap \text{Science})}{P(\text{Science})}$

Bayes' Theorem

Bayes' Theorem allows us to find the probability of an event given new information, especially when the direct conditional probability is not known.

Formula:

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

Example: Probability that a person has a disease given a positive test result, using test accuracy and disease prevalence.

Contingency Tables

A contingency table displays the frequency distribution of variables and is 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.

Example Table:

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

Marginal probability: $P(\text{Drives a Car}) = \frac{60}{100} = 0.6$

Joint probability: $P(\text{Senior} \cap \text{Drives a Car}) = \frac{40}{100} = 0.4$

Conditional probability: $P(\text{Drives a Car} | \text{Senior}) = \frac{40}{50} = 0.8$

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.

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 r objects from n is:

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

Example: Number of ways to arrange 3 out of 5 books:

$P(5, 3) = \frac{5!}{(5-3)!} = \frac{120}{2} = 60$

Permutations of Non-Distinct Objects

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

Formula:

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

Example: Arrangements of the word BANANA:

$\frac{6!}{3!2!1!} = \frac{720}{12} = 60$

Combinations

Combinations are selections of objects where order does not matter. The number of combinations of r objects from n is:

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

Example: Number of ways to choose 2 flavors from 5:

$C(5, 2) = \frac{5!}{3!2!} = 10$

Permutations vs. Combinations

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

Probability Using Counting Methods

Probability can be calculated using permutations and combinations when all outcomes are equally likely.

Formula:

$P(E) = \frac{\text{Number of favorable combinations}}{\text{Total number of combinations}}$

Example: Probability of winning a lottery by choosing 5 numbers out of 40:

$P(\text{win}) = \frac{1}{C(40, 5)}$