# Chapter 6 – Probability

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

Probability is “probably” one of the most routine calculative activities that we all human beings do. What’s the time? What’s the day? When would it be done? When would you be back home? The answer to all of these questions includes the concept of probability. It’s the probable thin line between “what it was” and “what it will be”. Mathematically, it deals with the possibilities of random situations or events. The value of probability ranges between zero to one. It is an essential topic in mathematics because it predicts the possibility of the occurrence of a random event.

The basic probability theory can be defined as the likelihood of occurrence of an event. This theory is also used in probability distribution to predict and learn about an experiment’s possible outcomes. The determination of a total number of outcomes is essential to find the probability of an event.

#### Probability Definition in Math

As you already know by now that probability is the likelihood of occurrence of an event. However, it is crucial to understand that not all probabilities can be predicted with certainty. It ranges from 0 to 1, 0 being an impossible event, and 1 means a certain event. In sample space, the probability of all events adds up to 1.

Let us consider an example for your better understanding; let’s say we toss a coin, it will either show up as a tail or a head, which means there would be only two certain outcomes (H, T). But what if we toss two or more coins together? The possibility increases. Let us assume two coins; there would be three possible outcomes, i.e., (H, H), (T, T), (H, T).

#### Formula for Probability

The formula for probability can be defined as the relationship between several favorable outcomes and the total number of outcomes. The ratio of both is defined as the formula for probability. #### Probability Tree

The probability tree is the visualization of different possible outcomes. The two main positions are the branches and end of the tree. The probability of an event is denoted from the branches and the outcomes from the ends of the tree. The probability tree helps to identify the pattern like when to multiply and when to add.

#### Types of Probability

The types of probability are categorized into three parts which are listed as follows:

1. Theoretical Probability
2. Experimental Probability
3. Axiomatic Probability

#### Theoretical Probability

Theoretical probability defines the reasoning behind the probability of an event. It identifies the possible outcomes of an event with reasoning. For example, the probability of getting a spade of ten is 1/52.

#### Experimental Probability

The next type is experimental probability, and the basis of experimental probability is observations of an experiment. The formula is the same as the original probability formula, i.e., the ratio between the number of favorable outcomes and a total number of outcomes. Let us take an example: in a deck of 52 cards, the probability of getting a ten is 4 divided by the total number of outcomes, i.e., 52. So the experimental probability would be 4/52.

#### Axiomatic Probability

The axiomatic Probability is based on the axiom rules. The axiomatic probability approach determines the occurrence or non-occurrence of the events. The chances of an event can be quantified with this approach. The axioms are applied to all sorts of types and events. Kolmogorov introduced the axioms, also called Kolmogorov’s three axioms. The three axioms are:

1. The probability of an event E will always be greater than or equal to zero. It can never be less than zero.
2. The probability of occurrence of sample space will be 1.
3. The probability of an event happening is equal to the sum of the probability of both the events for mutually exclusive events.

At MSVgo, you can learn the concept of Binomial Probability in an articulate step-by-step visualization followed by multiple examples of the topic.

#### Probability of an Event

Let us assume that an event E can occur in ‘r’ number of ways and the probability ‘p’ is equally likely. Then the probability of the occurrence of an event would be written as:

P(E) = r/n

The failure or the probability that doesn’t occur would be written as:

P(E’) = (n-r) / n = 1 – (r/n)

E’ is represented as an event that will not occur.

Therefore, it will be

P(E) + P(E’) = 1

It depicts the total possibilities of an event that will occur or not occur in an experiment and is equal to 1.

#### Equally Likely Events

The same theoretical probability of the events is known as equally likely events. Event is the single outcome of an experiment, and the outcome is the possible or probable result of an experiment, as you must have noticed in your day-to-day life. The results are equally likely if the occurrence of events all has the same probability. Let us consider an example; if you throw some dice, the probability of getting a 5 is 1/6. Likewise, the probability of getting the remaining numbers one by one would be 1/6. The example of equally likely events are as follows:

• Getting a 10 or 2 in a deck of cards
• Getting 1, 4 and 6 on a dice
• Getting a pink colored book from a stack of books

The probability of every event would have the same equally likely event probability.

#### Complementary Events

The complementary events are the probability, which has only two outcomes. For example, the probability of a person coming to your house would be either yes or no. It is known as complementary events. The opposite of a probability is a complement of an event. More examples of complementary events are as follows:

• Today, you will either go to a place or not
• You will win the lottery, or you won’t
• You will either pass this exam or not

#### Probability Density Function

The probability density function can be defined as the relationship between the density of continuous random variables and ranges of values. It determines and identifies the distribution of the existence of mean and deviation. The standard distribution is used to determine the database and statistics used for the science representation and real-valued variables if the distribution is unknown.

Using the MSVgo app that provides a step by step conceptual breakdown of topics, you can understand the concept of “Additional Rule of Probability” and “Types of Complementary Events of Probability” in a very visually creative manner. To download the app now and see for yourself, click here.

### High School Physics

• Alternating Current
• Atoms
• Communication Systems
• Current Electricity
• Dual nature of Radiation and Matter
• Electric Charges and Fields
• Electricity
• Electromagnetic Induction
• Electromagnetic Waves
• Electron Beams and Radioactivity
• Electrons and Photons
• Electrostatic Potential and Capacitance
• Fluid Pressure
• Force and Acceleration
• Force And Laws Of Motion
• Gravitation
• Internal Energy
• Kinetic Theory
• Law of motion
• Light – Reflection And Refraction
• Magnetic Effects Of Electric Current
• Magnetism and Matter
• Management Of Natural Resources
• Mechanical properties of Fluids
• Mechanical properties of Solids
• Motion
• Motion in a plane
• Motion in a straight line
• Moving Charges and Magnetism
• Nuclear Energy
• Nuclei
• Oscillations
• Our Environment
• Paths of Heat
• Physical world
• Ray optics and optical instruments
• Semiconductor Devices
• Semiconductor Electronics: Materials, Devices and Simple Circuits
• Simple Machines
• Sound
• Sources Of Energy
• Specific and Latent Heats
• Spherical Mirrors
• Static Electricity
• Systems of Particles and Rotational motion
• Thermal properties of matter
• Thermodynamics
• Units and Measurement
• Vectors, Scalar Quantities and Elementary Calculus
• Wave Optics
• Waves
• Work, Power and Energy

### High School Chemistry

• Acids, Bases and Salts
• Alcohols, Phenols and Ethers
• Aldehydes, Ketones and Carboxylic Acids
• Aliphatic and Aromatic Hydrocarbons
• Alkyl and Aryl Halides
• Amines
• Analytical Chemistry
• Atomic Structure
• Atoms And Molecules
• Basic concepts of Chemistry
• Biomolecules
• Carbon And Its Compounds
• Carboxylic acids and Acid Derivatives
• Chemical Bonding and Molecular Structures
• Chemical Energetics
• Chemical Equilibria
• Chemical Kinetics
• Chemical Reactions And Equations
• Chemical Reactions and Their Mechanisms
• Chemistry in Everyday Life
• Chemistry of p-Block elements
• Chemistry of Transition and Inner Transition
• Classification of Elements
• Coordination Compounds
• Cyanide, Isocyanide, Nitro compounds and Amines
• Electrochemistry
• Electrolysis
• Elements, Compounds and Mixtures
• Environmental Chemistry
• Equilibrium
• Ethers and Carbonyl compounds
• Haloalkanes and Haloarenes
• Hydrocarbons
• Hydrogen
• Ideal solutions
• Introduction to Organic Chemistry
• Ionic equilibria
• Matter
• Matter Around Us
• Matter In Our Surroundings
• Metallurgy
• Metals And Non-Metals
• Mole Concept and Stoichiometry
• Natural Resources
• Organic Chemistry – Basic Principles
• Periodic Classification of Elements
• Physical and Chemical Changes
• Physical and Chemical Properties of Water
• Polymers
• Preparation, Properties and Uses of Compounds
• Principles and Processes of Isolation of Elements
• Redox Reactions
• Relative Molecular Mass and Mole
• States of Matter
• Structure Of The Atom
• Study of Compounds
• Study of Gas Laws
• Study of Representative Elements
• Surface Chemistry
• The d-block and f-block elements
• The Gaseous State
• The p-Block Elements
• The Periodic Table
• The s-Block Elements
• The Solid State
• Thermodynamics

### High School Biology

• Absorption and Movement of Water in Plants
• Adolescent Issues
• Anatomy of Flowering Plants
• Animal Kingdom
• Bacteria and Fungi-Friends and Foe
• Biodiversity and Conservation
• Biofertilizers
• Biological Classification
• Biomedical Engineering
• Biomolecules
• Biotechnology and its Applications
• Biotic Community
• Body Fluids and Circulation
• Breathing and Exchange of Gases
• Cell – Unit of Life
• Cell Cycle and Cell Division
• Cell Division and Structure of Chromosomes
• Cell Reproduction
• Cellular Respiration
• Chemical Coordination and Integration
• Circulation
• Control And Coordination
• Crop Improvement
• Digestion and Absorption
• Diversity In Living Organisms
• Ecosystem
• Environmental Issues
• Excretory Products and their Elimination
• Flowering Plants
• Genes and Chromosomes
• Health and Diseases
• Health and Its Significance
• Heredity And Evolution
• Heredity and Variation
• How Do Organisms Reproduce?
• Human Diseases
• Human Eye And Colourful World
• Human Health and Disease
• Human Population
• Human Reproduction
• Hygiene
• Improvement In Food Resources
• Integumentary System- Skin
• Kingdom Fungi
• Kingdom Monera
• Kingdom Protista
• Life Processes
• Locomotion and Movement
• Microbes in Human Welfare
• Mineral Nutrition
• Molecular Basis of Inheritance
• Morphology of Flowering Plants
• Neural Control And Coordination
• Nutrition in Human Beings
• Organism and Population
• Photosynthesis
• Photosynthesis in Higher Plants
• Plant Growth and Development
• Plant Kingdom
• Pollination and Fertilization
• Pollution; Sources and its effects
• Principles of Inheritance and Variation
• Reproduction and Development in Angiosperms
• Reproduction in Organisms
• Reproductive Health
• Respiration in Human Beings
• Respiration in Plants
• Respiratory System
• Sexual Reproduction in Flowering Plants
• Strategies for Enhancement in Food Production
• Structural Organisation in Animals
• Structural Organisation of the Cell
• The Endocrine System
• The Fundamental Unit Of Life
• The Living World
• The Nervous System and Sense Organs
• Tissues
• Transpiration
• Transport in Plants

### High School Math

• Algebra – Arithmatic Progressions
• Algebra – Complex Numbers and Quadratic Equations
• Algebra – Linear Inequalities
• Algebra – Pair of Linear Equations in Two Variables
• Algebra – Polynomials
• Algebra – Principle of Mathematical Induction
• Algebra – Quadratic Equations
• Binomial Theorem
• Calculus – Applications of Derivatives
• Calculus – Applications of the Integrals
• Calculus – Continuity and Differentiability
• Calculus – Differential Equations
• Calculus – Integrals
• Geometry – Area
• Geometry – Circles
• Geometry – Conic Sections
• Geometry – Constructions
• Geometry – Introduction to Euclid’s Geometry
• Geometry – Three-dimensional Geometry
• Geometry – Lines and Angles
• Geometry – Quadrilaterals
• Geometry – Straight Lines
• Geometry – Triangles
• Linear Programming
• Matrices and Determinants
• Mensuration – Areas
• Mensuration – Surface Areas and Volumes
• Number Systems
• Number Systems – Real Numbers
• Permutations and Combinations
• Probability
• Sequence and Series
• Sets and Functions
• Statistics
• Trignometry – Height and Distance
• Trignometry – Identities
• Trignometry – Introduction

### Middle School Science

• Acids, Bases And Salts
• Air and Its Constituents
• Basic Biology
• Body Movements
• Carbon and Its Compounds
• Cell – Structure And Functions
• Changes Around Us
• Chemical Effects Of Electric Current
• Chemistry in Your Life
• Coal And Petroleum
• Combustion And Flame
• Components Of Food
• Conservation Of Plants And Animals
• Crop Production And Management
• Electric Current And Its Effects
• Electricity And Circuits
• Elements and Compounds
• Fibre To Fabric
• Food production and management
• Force And Pressure
• Forests: Our Lifeline
• Friction
• Fun With Magnets
• Garbage In, Garbage Out
• Getting To Know Plants
• Health and Hygiene
• Heat
• Hydrogen
• Life Processes: Nutrition in Animals and Plants
• Light, Shadows And Reflections
• Materials: Metals And Non-Metals
• Matter and Its States
• Metals and Non-metals
• Micro Organisms: Friend And Foe
• Motion And Measurement Of Distances
• Motion And Time
• Nutrition In Animals
• Nutrition In Plants
• Organization in Living Things
• Our Environment
• Physical And Chemical Changes
• Pollution and conservation
• Pollution Of Air And Water
• Reaching The Age Of Adolescence
• Reproduction In Animals
• Reproduction In Plants
• Respiration In Organisms
• Rocks and Minerals
• Separation Of Substances
• Simple Machines
• Soil
• Some Natural Phenomena
• Sorting Materials Into Groups
• Sound
• Stars And The Solar System
• Structure of Atom
• Synthetic Fibers And Plastics
• The Living Organisms And Their Surroundings
• Transfer of Heat
• Transformation of Substances
• Transportation In Animals And Plants
• Universe
• Waste-water Story
• Water: A Precious Resource
• Weather, Climate And Adaptations Of Animals To Climate
• Winds, Storms And Cyclones

### Middle School Math

• Addition
• Area and Its Boundary
• Boxes and Sketches
• Data Handling
• Fun With Numbers
• Heavy and Light
• How Many
• Long And Short
• Mapping
• Measurement
• Money
• Multiplication and Factors
• Multiply and Divide
• Numbers
• Parts and Wholes
• Pattern Recognition
• Patterns
• Play With Patterns
• Rupees And Paise
• Shapes And Angles
• Shapes And Designs
• Shapes and Space
• Similarity
• Smart Charts
• Squares
• Subtraction
• Tables And Shares
• Tenths and Hundredths
• Time
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