StudySmarter - The all-in-one study app.

4.8 • +11k Ratings

More than 3 Million Downloads

Free

Probability

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Point Estimation
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Standard Deviation
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Survey Bias
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

**Probability** is the branch of mathematics that studies the numerical description of how likely it is that an event will happen. Probability covers real-life situations where it's difficult to predict whether they will happen or not because their possible outcomes occur randomly. For example, when tossing a coin, you don't know if it will land on heads or tails. Despite this, you do have valuable information that can be used to predict how likely it is for each outcome to happen.

When talking about probability, there are a few concepts that you will need to familiarize yourself with:

An

**experiment**is a process that can be repeated many times, producing a set of specific outcomes, ie tossing a coin or rolling a die.An

**event**is the outcome or set of outcomes resulting from an experiment, ie when tossing a coin, a possible event will be getting tails; when rolling a die, an event will be getting a 4.The

**sample space**is the set of all possible outcomes, ie the sample space when tossing a coin is heads and tails, and the sample space when rolling a die is 1, 2, 3, 4, 5, and 6.The value that describes the probability of an event can range from 0 (zero) to 1.

A probability of 0 (zero) is considered an

**impossible**event.A probability of 1 refers to a

**certain**event.If the probability of an event is 0.5, then the event is

**equally likely**to happen as it is not to happen.Any event with a probability between 0 and 0.5 is considered unlikely to happen, and any event with a probability between 0.5 and 1 is considered likely to happen.

Please refer to the Events (Probability) article to learn more about the probability of different types of events.

The formula to calculate the probability of an event is as follows:

Probabilities can be expressed in fractions, decimals or percentages. For example, when tossing a coin, the probability of it landing on tails is , which is the same as saying 0.5 or 50%.

The main rules of probability that you need to keep in mind when calculating probabilities are:

The probability of an event happening ranges between 0 (zero) and 1:

The sum of the probabilities of all possible outcomes equals 1.

**Complement rule:**The probability that an event does not happen equals 1 minus the probability of the event happening:

**General addition rule:**The probability of A or B happening equals the probability of A plus the probability of B minus the probability of A and B happening together:

**Addition rule for mutually exclusive events:**Mutually exclusive events cannot happen at the same time. To calculate the probability of A or B happening in this case, we use the addition rule:

The rule changes because for mutually exclusive events,

**Multiplication rule for independent events:**Two events are independent when the occurrence of one event does not affect the probability of another one happening. If A and B are independent, the probability of A and B happening together equals the probability of A times the probability of B:

**Conditional probability:**In this case, the probability of an event will be affected by another event that has happened. The conditional probability of B given that A happened is:

The denominator will be the probability of the given event.

When solving probability problems, it can get very confusing to work out all the possible outcomes from an experiment. To make this task easier, you can use diagrams specially designed for this purpose to create visual representations of all the possible outcomes. These diagrams are the Venn diagram and the Tree diagram. Let's see when you need to use each one.

**Venn diagrams** are very useful when solving probability problems, as they help you represent events graphically. A rectangle is used to represent the sample space (S), and inside the rectangle, you can draw oval shapes representing each event. You can also include the frequencies or the probabilities of each event in the diagram. Let's see the most common scenarios that you can represent with Venn diagrams for two events, A and B:

1. **Event A and B:** In this case, both A and B occur, represented by the intersection of the two ovals.

2. **Event A or B:** In this case, A or B or both occur, represented by the union of the two ovals.

3. **Event not A:** In this case, A does not occur, and it represents the complement of A.

There are 30 students in a tutor group, 15 students are studying French, 12 are studying Spanish, and 5 are studying both languages. Draw a Venn diagram to represent this information.

A = students studying French

B = students studying Spanish

Include the frequency of the intersection first, then work out the other values around it.

There are 5 students studying both languages, which leaves you with 10 students studying only French and 7 students studying only Spanish. That means that the remaining 8 students are not studying any languages.

Please follow the link to the Venn Diagrams article to expand your knowledge about this topic.

**Tree diagrams **are especially useful to represent all the possible outcomes when you have two or more events happening one after the other. To create a tree diagram, draw a branch for each outcome in an event. Each branch should point to its corresponding outcome and include the probability of occurrence of each outcome.

Let's represent the possible outcomes when tossing a coin twice:

The sample space is where H = head and T = tail

If you go through each branch, all the possible outcomes are: HH, HT, TH and TT. The probability of the coin landing on H or T is every time, no matter how many times you toss the coin.

You can read the Tree Diagram article to expand your knowledge about this topic.

Here are a couple of examples of how to calculate probability:

Based on the information provided by the Venn Diagram that we created in the previous section:

A = students studying French

B = students studying Spanish

Calculate the probability that a student selected at random:

a) Studies French

b) Studies Spanish

c) Studies Spanish but not French

d) Does not study any languages

**Solutions:**

a)

b)

c)

d)

Based on the Tree diagram from the previous section, if you want to calculate the probability of getting two heads or two tails (HH or TT), you can proceed as follows:

1. Find the probability of getting two heads (HH). To do this, you need to multiply the probabilities along that branch.

2. Now, find the probability of getting two tails (TT).

3. To find the probability of HH or TT happening, you need to add their probabilities together.

For more examples, check out the Probability Calculations article.

As mentioned in the probability rules, conditional probability refers to the probability of an event happening, given that another event has happened. The conditional probability of B given that A happened is:

The denominator will be the probability of the given event.

In a group of students, if one is selected at random, the probability of them liking football is 60%. The probability that the selected student likes football and is a male is 40%. If a student who likes football is selected, what is the probability that the student is also male?

F = student likes football.

M = student is male

Please follow the link to the conditional probability article to learn more about this topic.

A **probability distribution** is a table or equation that associates each possible outcome of a random variable with its corresponding probabilities. A **random variable** is a variable whose value is defined by the outcome of a random experiment. A variable is **discrete** when it can only take certain numerical values within a given interval. A variable is **continuous** when it can take infinite values within an interval.

A **discrete probability distribution** lists all the probabilities for each outcome of the random variable using a table.

The probability that the random variable *X* takes a specific value *x* is written like this: *P (X = x)*

Let's consider the experiment of tossing a coin two times. The possible outcomes are: HH, HT, TH, and TT. If we say that the variable X = number of tails, we can see from the possible outcomes that the possible values of X are 0, 1 and 2.

Now you can represent the information above as a table:

x | 0 | 1 | 2 |

P (X = x) |

The sum of the probabilities of all the possible outcomes equals 1: ∑ *P (X = x) = 1* .

In this example: ✔

The probability distribution of a continuous random variable is represented by an equation. This equation is called the **probability density function**, which has the following characteristics:

for all values of x

The area under the curve of the function is equal to 1.

The Probability Distribution study set has more about this topic!

Probability is the branch of mathematics that studies the numerical description of how likely it is that an event will happen.

Probability covers real-life situations that are difficult to predict whether they will happen or not because their outcomes are random.

An experiment is a process that can be repeated many times, producing a set of specific outcomes, ie tossing a coin or rolling a die.

We can express probabilities in fractions, decimals or percentages.

Venn diagrams and Tree diagrams are useful to represent the possible outcomes of an experiment when solving probability problems.

Conditional probability refers to the probability of an event happening, given that another event has happened.

A probability distribution is a table or equation that associates each possible outcome of a random variable with its corresponding probabilities.

There are seven probability rules:

1. The probability of an event happening ranges between 0 (zero) and 1: 0 ≤ P(A) ≤ 1

2. The sum of the probabilities of all possible outcomes equals 1.

3. Complement rule: P(not A) = 1 - P(A)

4. General addition rule: P(A or B) = P(A) + P(B) -P(A and B)

5. Addition rule for mutually exclusive events: P(A or B) = P(A) + P(B)

6. Multiplication rule for independent events: P(A and B) = P(A) x P(B)

7. Conditional Probability: P(B|A) = P(A and B)/P(A)

More about Probability

Be perfectly prepared on time with an individual plan.

Test your knowledge with gamified quizzes.

Create and find flashcards in record time.

Create beautiful notes faster than ever before.

Have all your study materials in one place.

Upload unlimited documents and save them online.

Identify your study strength and weaknesses.

Set individual study goals and earn points reaching them.

Stop procrastinating with our study reminders.

Earn points, unlock badges and level up while studying.

Create flashcards in notes completely automatically.

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.