Probability Rules and Formulas

Notation

Symbol	Meaning
$S$	Sample space — the set of all possible outcomes
$A, B$	Events — subsets of the sample space
$P(A)$	Probability that event $A$ occurs
$A'$ or $\bar{A}$	Complement of $A$ — the event that $A$ does not occur
$A \cap B$	Intersection — both $A$ and $B$ occur
$A \cup B$	Union — $A$ or $B$ (or both) occur
$P(A \mid B)$	Conditional probability — probability of $A$ given $B$ has occurred
$n(A)$	Number of outcomes in event $A$

Fundamental Rules

Rule	Formula	When to Use
Probability of an event	$P(A) = \dfrac{n(A)}{n(S)}$	Equally likely outcomes
Probability bounds	$0 \leq P(A) \leq 1$	Always true
Certain event	$P(S) = 1$	Something must happen
Impossible event	$P(\emptyset) = 0$	Cannot happen
Complement rule	$P(A') = 1 - P(A)$	Easier to find what does NOT happen

Addition Rules (OR)

Situation	Formula	Example
Mutually exclusive events	$P(A \text{ or } B) = P(A) + P(B)$	Rolling a 2 or a 5 on one die
General (overlapping events)	$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$	Drawing a red card or a face card

Events $A$ and $B$ are mutually exclusive if they cannot occur at the same time: $P(A \text{ and } B) = 0$ .

Multiplication Rules (AND)

Situation	Formula	Example
Independent events	$P(A \text{ and } B) = P(A) \cdot P(B)$	Flipping heads twice in a row
Dependent events	$P(A \text{ and } B) = P(A) \cdot P(B \mid A)$	Drawing two aces without replacement

Events $A$ and $B$ are independent if the occurrence of one does not affect the other: $P(B \mid A) = P(B)$ .

Conditional Probability

Formula	Use
$P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)}$	Probability of $A$ given $B$ has occurred
$P(B \mid A) = \dfrac{P(A \text{ and } B)}{P(A)}$	Probability of $B$ given $A$ has occurred

Testing for Independence

Two events are independent if and only if any one of these is true:

Test
$P(A \mid B) = P(A)$
$P(B \mid A) = P(B)$
$P(A \text{ and } B) = P(A) \cdot P(B)$

Bayes’ Theorem

$P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B)}$

Expanded form (when $B$ can occur through $A$ or $A'$ ):

$P(A \mid B) = \frac{P(B \mid A) \cdot P(A)}{P(B \mid A) \cdot P(A) + P(B \mid A') \cdot P(A')}$

Common application: A diagnostic test has sensitivity $P(+ \mid \text{disease})$ and specificity $P(- \mid \text{no disease})$ . Bayes’ theorem finds $P(\text{disease} \mid +)$ .

Counting Techniques

Method	Formula	Use
Fundamental counting principle	$n_1 \times n_2 \times \cdots \times n_k$	Total outcomes from $k$ sequential choices
Factorial	$n! = n(n-1)(n-2) \cdots (1)$	Arrangements of $n$ distinct items
	$0! = 1$ by definition
Permutations	$P(n, r) = \dfrac{n!}{(n-r)!}$	Ordered arrangements of $r$ from $n$
Combinations	$C(n, r) = \dbinom{n}{r} = \dfrac{n!}{r!(n-r)!}$	Unordered selections of $r$ from $n$

When to Use Permutations vs Combinations

	Order Matters	Order Does Not Matter
Choosing $r$ from $n$	Permutation: $P(n, r)$	Combination: $C(n, r)$
Example	Picking 1st, 2nd, 3rd place	Choosing a committee of 3

”At Least One” Shortcut

$P(\text{at least one}) = 1 - P(\text{none})$

This is often much simpler than adding up every possible case individually.

Common Discrete Distributions

Distribution	PMF	Mean	Std Dev
Binomial	$P(X = k) = \dbinom{n}{k} p^k (1-p)^{n-k}$	$\mu = np$	$\sigma = \sqrt{np(1-p)}$
Geometric	$P(X = k) = (1-p)^{k-1} p$	$\mu = \dfrac{1}{p}$	$\sigma = \dfrac{\sqrt{1-p}}{p}$
Poisson	$P(X = k) = \dfrac{e^{-\lambda} \lambda^k}{k!}$	$\mu = \lambda$	$\sigma = \sqrt{\lambda}$

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