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Probability Basics: Concepts, Rules, and Distributions

Key Terms
Main Concepts
Set Theory & Event Algebra
Basic Axioms of Probability
Rules of Probability
Combinatorial Probability
Random Variables and Distributions
Conditional Probability & Independence
Probability Function
Common Pitfalls
Probability Symbols Reference
Probability Basics FAQ











Introduction to Probability Section

Probability is a field of mathematics that deals with uncertainty and provides tools to measure and analyze how likely events are to occur. It begins with basic concepts such as outcomes, events, and sample spaces, forming the foundation for calculating likelihoods.
Central to probability is the concept of probability measures, which assign values between 0 and 1 to events, indicating their likelihood. An impossible event has probability 0, and a certain event has probability 1. Key principles include independence (events that do not influence each other) and conditional probability, which considers the likelihood of an event given that another has occurred.
Probability also introduces random variables, which assign numerical values to outcomes. These variables are categorized as either discrete (taking specific values, like rolling a dice) or continuous(taking any value within a range, like measuring temperature). Important measures such as expectancy(average value) and variance(spread or variability) are used to summarize the behavior of random variables.
Advanced topics include distributions, such as the binomial, normal, and Poisson distributions, which model specific types of random phenomena. These tools are essential for understanding patterns in random processes and making informed predictions.
Probability is widely applied in science, engineering, finance, and everyday decision-making. It forms the basis for statistics, enabling data-driven insights and predictions, and supports fields like machine learning, risk analysis, and quantum mechanics. By studying probability, students develop skills to reason about uncertainty and draw conclusions from incomplete information.

Key Terms

Foundations

Probability— a number in [0,1][0,1] assigned to each event
Random Experiment— a process whose outcome is uncertain
Sample Space— the set Ω\Omega of all possible outcomes
Event— a subset A⊆ΩA \subseteq \Omega of the sample space
Elementary Event— an event containing exactly one outcome
Relative Frequency— proportion of times an event occurs in nn trials
Probability Measure— the function PP assigning values in [0,1][0,1] to events
Equally Likely Events— events each assigned equal probability

Conditional Probability & Independence

Conditional Probability— P(A∣B)P(A \mid B), probability of AA given BB occurred
Independent Events— events where P(A∩B)=P(A)⋅P(B)P(A \cap B) = P(A) \cdot P(B)
Mutual Exclusiveness— events with A∩B=∅A \cap B = \emptyset

Random Variables

Random Variable— a function X:Ω→RX: \Omega \to \mathbb{R}
Discrete Random Variable— a random variable with countable range
Continuous Random Variable— a random variable taking values in an interval
Bernoulli Experiment— a two-outcome experiment: success or failure
Sequence of Bernoulli Trials— repeated independent Bernoulli experiments with constant pp
Function of a Random Variable— a new variable Y=g(X)Y = g(X)

Distribution Functions

Probability Mass Function— pX(x)=P(X=x)p_X(x) = P(X = x) for discrete variables
Probability Density Function— fX(x)f_X(x) such that P(a≤X≤b)=∫abfX(x) dxP(a \leq X \leq b) = \int_a^b f_X(x)\,dx
Cumulative Distribution Function— FX(x)=P(X≤x)F_X(x) = P(X \leq x)
PDF of a Transformed Variable— density of Y=g(X)Y = g(X) derived from density of XX

Measures

Expected Value— E[X]E[X], the weighted average of all values of XX
Variance— Var⁡(X)=E[(X−μ)2]\operatorname{Var}(X) = E[(X-\mu)^2], spread around the mean
Standard Deviation— σ=Var⁡(X)\sigma = \sqrt{\operatorname{Var}(X)}
Covariance— Cov⁡(X,Y)\operatorname{Cov}(X,Y), joint variability of two random variables
Correlation Coefficient— ρXY\rho_{XY}, normalized linear dependence
Conditional Expectation— E[X∣Y=y]E[X \mid Y = y], expected value given information
Conditional Variance— Var⁡(X∣Y=y)\operatorname{Var}(X \mid Y = y), spread given information
Moment— E[Xk]E[X^k], a numerical summary of distribution shape
Moment Generating Function— MX(t)=E[etX]M_X(t) = E[e^{tX}], encodes all moments

Discrete Distributions

Bernoulli Distribution— distribution for a single success/failure trial
Binomial Distribution— counts successes in nn independent Bernoulli trials
Geometric Distribution— number of trials until first success
Negative Binomial Distribution— trials needed to achieve rr successes
Poisson Distribution— counts events in a fixed interval at constant rate λ\lambda
Discrete Uniform Distribution— equal probability across a finite set of values
Hypergeometric Distribution— successes in nn draws without replacement

Continuous Distributions

Normal Distribution— symmetric bell-shaped distribution with parameters μ\mu, σ2\sigma^2
Exponential Distribution— waiting time between events at constant rate λ\lambda

Multivariate Probability

Bivariate Random Variable— a pair (X,Y)(X, Y) considered jointly
N-Variate Random Variables— a vector (X1,…,Xn)(X_1, \ldots, X_n) of random variables
Joint PMF— pX,Y(x,y)=P(X=x,Y=y)p_{X,Y}(x,y) = P(X = x, Y = y)
Joint PDF— fX,Y(x,y)f_{X,Y}(x,y), joint density for continuous variables
Joint CDF— FX,Y(x,y)=P(X≤x,Y≤y)F_{X,Y}(x,y) = P(X \leq x, Y \leq y)
Marginal Distribution— distribution of one variable from a joint distribution
Conditional PMF— pX∣Y(x∣y)p_{X|Y}(x \mid y) for discrete variables
Conditional PDF— fX∣Y(x∣y)f_{X|Y}(x \mid y) for continuous variables
Independent Random Variables— variables where joint distribution factorizes
Uncorrelated Random Variables— variables with Cov⁡(X,Y)=0\operatorname{Cov}(X,Y) = 0
Orthogonal Random Variables— variables with E[XY]=0E[XY] = 0

Set Operations

Union of Sets— A∪BA \cup B, all outcomes in AA or BB
Intersection of Sets— A∩BA \cap B, outcomes in both AA and BB
Complement of a Set— AcA^c, all outcomes in Ω\Omega not in AA
Disjoint Sets— sets with A∩B=∅A \cap B = \emptyset
Null Set— ∅\emptyset, the empty set representing an impossible event
Venn Diagram— graphical representation of sets as overlapping circles

Visual Tools

Probability Tree— a branching diagram for sequential random processes

See All Probability Definitions →


Main Concepts

The Sample Space (Ω), is the collection of all different results that the experiment may produce.
The sample space can be finite (for example, in the dice‐rolling scenario or coin flipping) or infinite (for instance, selecting a real number).
In addition, we may divide sample spaces by outcome types into discrete or continuous.
Often times, defining or calculating proper sample space for any given case may pose serious challenge and demands experience and certain analytic skills.
Although in the case of a dice roll the collection of possible outcomes may seem self‐evident, the sample space plays an important role in conducting more complex experiments. Typically, a researcher will take the sample space and partition it into subsets in order to draw various conclusions.
In any practical application, accurately defining the sample space is essential to solving probability problems.

In probability theory, the objects to which probabilities are assigned are called events.
By definition, an event is any subset of the sample space. This includes single outcomes, collections of outcomes, the empty set, and the entire sample space itself.
Probability Event is simply any subset of the sample space.
Example
In case of dice roll the sample space would be S={1,2,3,4,5,6}S = \{1, 2, 3, 4, 5, 6\}
Some possible events:
Event A={2,4,6}A = \{2, 4, 6\} (rolling an even number)
Event B={5}B = \{5\} (rolling exactly 5)
Event C={1,2,3,4,5,6}C = \{1, 2, 3, 4, 5, 6\} (any outcome - certain event)
Event D={}D = \{\} (impossible event)
As the definition states and the example shows, probability event may include one or more outcomes. It is a set of results counting as one event.
Probability is a function that assigns to each event in the sample space a real number in [0,1][0,1] where total probability value of the entire sample space P(S)=1P(S)=1.
This number is calculated as a ratio P(E)=Number of favorable outcomes for event ETotal number of possible outcomes in the sample space SP(E) = \frac{\text{Number of favorable outcomes for event E}}{\text{Total number of possible outcomes in the sample space S}}, when all outcomes in SS are equally likely
Probability function satisfies three basic axioms of probability.

Pitfall: Treating unequal outcomes as equally likely
Taking the eleven possible sums 2,3,…,122, 3, \ldots, 12 of two dice as the sample space gives P(sum=7)=111P(\text{sum} = 7) = \frac{1}{11}. The sums are not equally likely; counting the 3636 ordered pairs of faces gives 636=16\frac{6}{36} = \frac{1}{6}.

The ratio counts outcomes, so its result depends on which list of outcomes is taken as the sample space. For a single die the six faces are the natural list. For two dice there is a choice between the 3636 ordered pairs of faces and the 1111 possible sums, and the two lists give different answers to the same question.

Set Theory & Event Algebra

When we conceptualize probability, we naturally think of sample spaces as collections of individual outcomes—dots scattered across our mathematical landscape.
Sample Space Ω (Sample Space) Each dot represents an elementary event (outcome) Ω = {ω₁, ω₂, ω₃, ..., ωₙ} where each ωᵢ is an elementary event The sample space contains all possible outcomes of an experiment
However, this intuitive picture presents a fundamental challenge: if we treat each outcome as a geometric point, it has zero area by definition.
Consider the classical probability formula:
P(E)=Number of favorable outcomes for event ETotal number of possible outcomes in the sample space SP(E) = \frac{\text{Number of favorable outcomes for event E}}{\text{Total number of possible outcomes in the sample space S}}.
It gives each of the nn outcomes a mass of 1n\frac{1}{n}. If instead we treated outcomes as geometric points (dots), each with zero "probability mass," we'd face the paradox that every single outcome has probability zero, yet their sum must equal one.
The Zero Probability Paradox Why Individual Outcomes Have Zero Probability Ω (Sample Space) Single outcome ωᵢ Area = 0 Classical Probability Formula: P(A) = |A| / |Ω| For a single outcome ωᵢ: P({ωᵢ}) = 1 / |Ω| But if each dot has zero area: P({ωᵢ}) = 0 / ∞ = 0 The Paradox: • Each individual outcome has probability 0 • Yet the sum of all individual probabilities must equal 1 • This contradiction shows we cannot work with individual points! Solution: Work with Sets of Outcomes Event A (Set of outcomes) P(A) = |A| / |Ω| > 0 Now we get meaningful probabilities!

Fact: Random numbers are almost never fractions
Picked uniformly from [0,1][0,1], a number is a fraction with probability 00: the fractions can be listed one by one, and countably many zeros add up to 00. Additivity stops at countable sums, so the uncountably many points of [0,1][0,1] can still carry a total of 11.

This apparent contradiction reveals why probability theory fundamentally operates with sets rather than isolated points. We don't manipulate individual outcomes; instead, we work with collections or groups of outcomes. An event isn't a single dot—it's a set of possible outcomes that satisfy our condition of interest.

This set-theoretic foundation makes perfect sense: when we ask "what's the probability of rolling an even number on a die," we're really asking about the set {2,4,6} \{2, 4, 6\}, not about individual outcomes in isolation.

By treating events as sets, we gain access to the full power of set theory and algebra of sets laws for probability calculations. This mathematical framework provides elegant tools for combining and manipulating events—operations like union and intersection become natural ways to express complex probabilistic relationships, while concepts such as subsets and complements offer systematic approaches to analyzing event dependencies and exclusions.
Ω (Sample Space) A Legend: Sample Space (Ω) Event A Key Insight: The circle represents event A as a set of outcomes. Area of A P(A) = Area of Ω Probability as ratio of areas Events as Sets in Probability Theory
To visualize these relationships between events-as-sets, we use Venn diagrams—powerful tools that illustrate unions, intersections, complements, and other set operations that form the algebraic backbone of probability theory.
Probability lives on sets of outcomes, not on single points, so the algebra of sets becomes the algebra of events.

Basic Axioms of Probability

The three Kolmogorov axioms provide a minimal yet complete framework for assigning consistent probabilities to events, laying the groundwork for all of probability theory. From these principles flow essential rules—such as the addition rule for disjoint events and Bayes’ theorem—as well as many useful corollaries that drive rigorous problem‐solving in statistics, science, and engineering.
  • •
    Non-negativity axiom
    For any event A, P(A) ≥ 0 (together with the other two axioms this gives P(A) ≤ 1).
  • •
    Normalization axiom
    P(S)=1P(S) = 1, meaning the whole sample space S has probability exactly 1
  • •
    Countable additivity axiom
    If A₁, A₂, … are disjoint, then P(⋃ᵢ Aᵢ) = ∑ᵢ P(Aᵢ).
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Rules of Probability

Probability rules translate the axioms of probability into practical tools for quantifying uncertainty. By systematically combining events—through complements, unions, intersections, and conditioning—they form the backbone of both classical (combinatorial) analyses and more advanced topics.
With these rules in hand, you’re ready to tackle sections on combinatorial models, discrete and continuous distributions, conditional probability, Bayesian inference, and beyond. Refer back to our overall probability breakdown to see how each subtopic weaves together in your learning journey.
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Combinatorial Probability


Fact: Probability began with an unfinished game
In 1654 Pascal and Fermat asked how to split the stakes of a game between two equal players, stopped when one needed 22 more wins and the other 33. Fermat imagined all 44 possible remaining rounds played out, even past the deciding one, so that the 1616 win-loss sequences are equally likely; 1111 favor the first player, so the stakes split 11:511 : 5.

Counting is the oldest tool in probability. Long before axioms or distributions, questions about games of chance were settled by listing the equally likely ways a game could unfold and dividing the favorable cases by the total. The hard part was rarely the division; it was finding a list of cases that really are equally likely.

Why Combinatorial Counting Remains Essential
Even with powerful general tools—probability distributions, conditional‐probability identities, and set algebra theorems—the direct application of combinatorial principles is often the most effective method:

Fundamental Combinatorial Rules
Employ the basic counting principle, permutations P(n,k)P(n,k), combinations (nk)\binom{n}{k}, and related identities (e.g. the binomial coefficient) to enumerate equally likely outcomes directly.

Simplicity
When all outcomes share equal likelihood, computing (nk)\binom{n}{k} or n×(n−1)×⋯n\times(n-1)\times\cdots is more straightforward than constructing full distribution tables or applying Bayes’s theorem.

Transparency
A step-by-step combinatorial argument—through case analysis or symmetry—makes explicit how each arrangement or selection contributes to the overall probability, avoiding opaque algebraic manipulation.

Efficiency for Small Sample Spaces
In problems involving a modest number of cards, dice, or slots, direct computation of permutations or combinations typically requires fewer conceptual steps than invoking general-purpose formulas.

Conceptual Insight
Deriving results via combinatorial identities deepens understanding of why certain events are more prevalent, reinforcing intuition that may be obscured by formulaic approaches.

Problem-Specific Customization
Combinatorics allows tailored strategies—case distinctions, bijective mappings, or the inclusion–exclusion principle—adapted to a problem’s unique constraints, rather than forcing it into a universal template.

Random Variables and Distributions

As we defined earlier, sample space SS is the full list of “all that can happen” in a given experiment.
But are all outcomes equally likely?
The answer is: it depends.
As we know from everyday experience, some experiments—like flipping a fair coin or rolling a fair die—assign the same probability to each outcome, while in others certain outcomes carry more weight.
To capture how those weights are assigned—and how they change when we look at different features of the same experiment—we need the formal notion of a probability distribution.
In many problems, interest centers not on the raw outcomes themselves but on some numerical feature of those outcomes—what we call a random variable.
A Random Variable is simply a rule that assigns a number to every elementary outcome in the sample space.
By doing this, it becomes possible to talk about averages, variances and more, using the full toolbox of arithmetic and calculus.
The Probability Distribution of a random variable then tells us how likely each numerical value is to happen.
It does this by gathering together all the elementary outcomes that map to the same number (or fall into the same range) and adding up their probabilities. Even when every outcome in the sample space is equally likely—say, each face of a fair die—different choices of random variable (for example, the face value itself versus “even or odd,” or “number of sixes in two rolls”) will group those outcomes differently. As a result, each of those measurements has its own distinct distribution, reflecting the particular way it “reads” the experiment.
At its heart, working with a probability distribution is simply about deciding how to spread out your “degree of belief” over every thing that could happen, and then using that spread to answer questions about uncertainty.

• Assigning weight: You begin by giving each possible outcome a nonnegative number (its weight), in such a way that all the weights add up to one.
• Capturing uncertainty: Those weights encode exactly how confident you are in each outcome, from “almost impossible” to “almost certain.”
• Calculating what matters: Once the weights are set, you can systematically compute things like “how much total weight falls in this region of outcomes,” or “what’s the average we’d expect,” or “how wildly outcomes vary.”
• Guiding decisions: With those calculations in hand, you can compare different spreads of belief, choose actions that maximize your expected gain, or measure how risky a plan is.
There are many different probability distributions—each with its own characteristic pattern—but they can be broadly classified into two main categories: discrete distributions, which assign probabilities to countable outcomes, and continuous distributions, which use density functions over intervals of real numbers.
Discrete Distributions Continuous Distributions Used for countable data Used for measurable data
One experiment can carry many distributions: each random variable regroups the same total probability of 1 by its own values.
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Conditional Probability & Independence

Conditional probability is simply the chance of something happening once you already know something else has happened. It tells you how your outlook on an event shifts when you gain new information about another event's occurrence.
When you learn about conditional probability, you’re really seeing how knowing that one event happened (B) changes your “bet” on another event (A). That change (or lack of change) is exactly what we mean by dependence or independence:

Dependent events:
If knowing that B occurred does change your opinion about AA, then AA and BB are dependent.
P(A∣B)  ≠  P(A).P(A\mid B)\;\neq\;P(A).
Or in simple words: “Once I see BB happen, my chance of AA goes up or down compared to what I thought before.”

Independent events:
If knowing that BB occurred doesn’t change your opinion about likelihood of AA, then AA and BB are independent.
P(A∣B)  =  P(A)P(A\mid B)\;=\;P(A)
Equivalently, the fact that BB happened gives you zero new information about AA.

The multiplication rule for independent events is:
P(A∩B)  =  P(A) P(B)P(A\cap B)\;=\;P(A)\,P(B)
The intuition behind it:
If two events don’t influence each other, the probability that both happen is just the product of their individual chances. In other words, to find the chance of A and B occurring together, you multiply P(A) by P(B).

Pitfall: Multiplying dependent events
For one die roll, with AA = even and BB = greater than 33, the product gives P(A∩B)=12⋅12=14P(A \cap B) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}. But A∩B={4,6}A \cap B = \{4, 6\}, so P(A∩B)=26=13P(A \cap B) = \frac{2}{6} = \frac{1}{3}: the events are dependent.

The product formula rests on the condition that knowing BB leaves the chance of AA unchanged. That condition is a property of the particular experiment and the particular pair of events, and it can be tested directly by comparing P(A∣B)P(A \mid B) with P(A)P(A). When the two differ, P(A∩B)P(A \cap B) is found from the conditional probability instead, as P(B) P(A∣B)P(B)\,P(A \mid B).

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Probability Function

Almost every probability topic ultimately relies on a single idea:
A probability function describes how probability is distributed across all possible outcomes of a random variable.
This rule is called a probability function, and it is the mathematical relation that turns a vague notion of uncertainty into something precise and analyzable.

A probability function specifies how likely each possible value of a random variable is.
It determines how probability is distributed across the outcomes and forms the basis on which all familiar probability distributions are built. Whether we study simple models like coin tossing or more structured distributions that arise in statistics, the probability function is always the mechanism operating underneath.
The Central Role of the Probability Function Random Experiment (vague uncertainty) "How likely?" Probability Function P(X = x) The mathematical rule that assigns probabilities to outcomes x₁ P(x₁) x₂ P(x₂) xₙ P(xₙ) Outcome 1 Outcome 2 Outcome n Forms the basis of... Probability Distributions Binomial, Poisson, Normal, etc. Key Properties: • 0 ≤ P(x) ≤ 1 • ΣP(x) = 1 • Precise & analyzable Transforms: Vague uncertainty ↓ Mathematical precision
Because of its central role, this concept receives its own dedicated page, where the idea is developed more formally and connected to the structure of probability distributions.
If you want to understand what a distribution really is, the probability function is the natural place to begin.
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Common Pitfalls

Each of the two mistakes below is marked where it happens in the sections above; here each is worked through in full, with the wrong and the right working side by side.

Treating unequal outcomes as equally likely

The ratio of favorable outcomes to all outcomes is applied to a list of outcomes that are not equally likely. The ratio counts outcomes, so it gives the right probability only when every outcome on the list carries the same weight.

Wrong working: two dice are rolled, and the sample space is taken as the possible sums 2,3,…,122, 3, \ldots, 12. That is 1111 outcomes, one of them favorable:
P(sum=7)=111P(\text{sum} = 7) = \frac{1}{11}


Right working: the equally likely outcomes are the 3636 ordered pairs (first die, second die). Six of them give a sum of 77: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1)(1,6), (2,5), (3,4), (4,3), (5,2), (6,1).
P(sum=7)=636=16P(\text{sum} = 7) = \frac{6}{36} = \frac{1}{6}


Why it happens: the question is about the sum, so the sums look like the natural outcomes. The list of sums does not show that different sums arise from different numbers of pairs.

How to check: compare two outcomes on the list. A sum of 22 arises only from (1,1)(1,1), with probability 136\frac{1}{36}, while a sum of 77 arises in six ways; the two cannot have the same probability.

Multiplying dependent events

The probability that two events both happen is found as P(A) P(B)P(A)\,P(B) without checking that the events are independent. The product formula holds only when knowing one event leaves the chance of the other unchanged.

Wrong working: one roll of a fair die; AA = even ={2,4,6}= \{2, 4, 6\}, BB = greater than 33 ={4,5,6}= \{4, 5, 6\}.
P(A∩B)=P(A) P(B)=12⋅12=14P(A \cap B) = P(A)\,P(B) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}


Right working: A∩B={4,6}A \cap B = \{4, 6\}, two of the six faces.
P(A∩B)=26=13P(A \cap B) = \frac{2}{6} = \frac{1}{3}

The same result comes from the conditional probability: P(B) P(A∣B)=12⋅23=13P(B)\,P(A \mid B) = \frac{1}{2} \cdot \frac{2}{3} = \frac{1}{3}.

Why it happens: the product rule is short and is often the first rule met for "and", so it is applied by habit, and independence is assumed instead of checked.

How to check: with six equally likely faces, every probability for one roll is a whole number of sixths, and 14\frac{1}{4} is not. Comparing P(A∣B)=2/63/6=23P(A \mid B) = \frac{2/6}{3/6} = \frac{2}{3} with P(A)=12P(A) = \frac{1}{2} shows the events are dependent.

Probability Symbols Reference

Our Probability Symbols page delivers a systematic reference for notation used in probability theory and statistics. This collection serves as an essential guide for students and professionals working with statistical concepts.
The reference organizes symbols into practical categories including probability notations (P(A), P(A|B)), random variables and distributions (f_X(x), F_X(x)), and common distribution families (Bin(n,p), N(μ,σ²)). It extends to advanced topics like statistical measures (E(X), Var(X)), hypothesis testing parameters (H₀, α, p-value), and information theory metrics (H(X), I(X;Y)).
Specialized sections cover moment generating functions (M_X(t)), key probability inequalities (Markov's, Chebyshev's), Bayesian methods, and regression analysis notation—all presented with precise LaTeX formatting to support academic writing and research in probability and statistics.
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Probability Basics FAQ

How did Pascal and Fermat split the stakes of an unfinished game?

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By counting equally likely endings. One player needed 22 more wins and the other 33, so at most 44 more rounds would settle the game. Fermat imagined all 44 rounds played out, even past the deciding one, which makes the 24=162^4 = 16 win-loss sequences equally likely. The first player wins at least 22 of the 44 rounds in 6+4+1=116 + 4 + 1 = 11 of them, so the stakes split 11:511 : 5. This exchange of letters in 1654 is often taken as the start of probability theory.Read more →

When is counting outcomes easier than using a probability formula?

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When the outcomes are equally likely and there are not too many of them. For a few cards, dice or slots, counting with permutations P(n,k)P(n,k) and combinations (nk)\binom{n}{k} usually takes fewer steps than building a distribution table or applying the Bayes theorem, and it shows where each arrangement comes from. The method rests on one condition: the list being counted must really consist of equally likely cases, which is where most counting mistakes start.Read more →

What is the chance that a random number between 0 and 1 is a fraction?

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Zero. The fractions in [0,1][0,1] can be listed one by one, each single number has probability 00 when the number is picked uniformly, and countable additivity adds countably many zeros to 00. So the number is irrational with probability 11. This is no contradiction: additivity covers only countable sums, and the uncountably many points of [0,1][0,1] can together carry a total of 11 while each carries nothing. That is why probability is assigned to sets of outcomes rather than to single points.Read more →

Can you find P(A and B) by multiplying P(A) and P(B)?

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Only when AA and BB are independent. On one roll of a fair die, take AA = even and BB = greater than 33. The product gives 12⋅12=14\frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}, but both happen only on 44 and 66, so P(A∩B)=26=13P(A \cap B) = \frac{2}{6} = \frac{1}{3}. A quick check: every probability for one roll is a whole number of sixths, and 14\frac{1}{4} is not. In general use P(A∩B)=P(B) P(A∣B)P(A \cap B) = P(B)\,P(A \mid B), here 12⋅23=13\frac{1}{2} \cdot \frac{2}{3} = \frac{1}{3}.Read more →

What is the difference between uncorrelated and orthogonal random variables?

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Uncorrelated means Cov⁡(X,Y)=0\operatorname{Cov}(X,Y) = 0, that is E[XY]=E[X] E[Y]E[XY] = E[X]\,E[Y]; orthogonal means E[XY]=0E[XY] = 0. Since Cov⁡(X,Y)=E[XY]−E[X] E[Y]\operatorname{Cov}(X,Y) = E[XY] - E[X]\,E[Y], the two conditions are the same when at least one of the variables has mean 00; otherwise no pair can satisfy both. Two independent fair dice are uncorrelated, yet E[XY]=3.5⋅3.5=12.25E[XY] = 3.5 \cdot 3.5 = 12.25, so they are not orthogonal.Read more →