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Chapter 7 – Probability

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

Introduction

The definition of probability states that the number of possible outcomes of any events is called its probability. It is the branch of math that deals with possibilities of results while doing random experiments. The value of each result is expressed between 0 and 1. Probability was introduced to determine the outcome of many worldly events and situations.

According to mathematics, probability is defined as the possibility of any event happening. Several events can’t be predicted with utmost certainty, but the formulas and theories of probability help us reach the outcome. It helps in predicting the chances of any event from happening. In this chapter, students are taught the simple problems on single events initially, but as the topics progress, you get to learn more challenging things. 

The probability of any event cannot be less than 0 or more than 1. It means the possibility of any event from happening will never be less than 0 and cannot be more than 1. It can also be said that all the sample spaces of an event will always add up to 1.

It is easy to find the probability of any event when simple problems on single events are given. But on the other hand, here is what you can do when multiple events are given. 

Probability of event to happen P(E) = Number of favourable outcomes/Total Number of outcomes

It is the basic formula that helps determine an event’s probability, but other formulas are used in different situations. People usually confuse between desirable outcomes and favourable outcomes.

Probability is categorised into three major types:

  • Theoretical probability

The theoretical probability can be defined as possible outcomes of any event to happen. This type of probability depends mainly on the reasoning behind the concept of probability. For example, when you toss a coin, the probability of getting a tail or head is fixed, which is ½ or 50%. 

  • Experimental probability

As the name suggests, this is the category where the probability of any event to happen is based on an experiment’s observations. The experimental probability can be determined by dividing the number of possible outcomes by the total number of outcomes. 

  • Axiomatic probability  

The axiomatic probability is the one that is governed by a specific set of rules or so-called axioms which apply to all types. These axioms are popularly known as Kolmogorov’s three axioms.

Let’s assume that an event ‘E’ can occur in ‘r’ ways and the total number of outcomes be ‘n’. The equation of probability can be expressed as:

P(E)= r/n

The probability of the event not occurring is expressed as:

P(E’)= (n-r) /n = 1- (r/n), where E’ is the probability of event not happening

We can also say, 

P(E)+ P(E’)= 1

  • Sample space
    It is defined as the set of possible outcomes that occur in an event.
  • Sample point
    It is defined as one of the possible results of an event.
  • Trial or experiment
    It is an event or trial where the outcomes are uncertain and only predicted.
  • Event
    It is defined as the single outcome of any experiment.
  • Outcome
    It is defined as the possible or probable result of any event or experiment.
  • Complimentary event
    It is defined as the non-happening events in the experiment.

Example 1: What will be the probability of getting four on rolling a die.

Ans:

Sample Space = {1, 2, 3, 4, 5, 6}

Number of favourable event = 1

i.e. {4}

Total number of outcomes = 6

Thus, Probability, P = ⅙.

Example 2: A bag contains five blue balls, four red balls and 11 white balls. If you draw out three balls randomly from the bag, what is the probability that the first ball will be red, the second ball would be blue, and the third ball would be white?

Ans:

The probability of getting the first ball being red is 4/20.

Now, 

We have drawn a ball for the first event to occur, which means the number of possibilities left for the second event will be 20 – 1 = 19.

Therefore, the probability of getting the second ball being blue is 4/19.

With two events already done, the number of possibilities left for the third event is 19 – 1 = 18.

The probability of the third ball being white is 11/18.

Therefore, the probability is 5/20 x 4/19 x 11/18 = 44/1368 = 0.032.

Example 3: A coin is tossed thrice. What is the probability of getting at least one head?

Ans: 

Sample space or total number of possible outcomes= [HHH, HHT, HTH, THH, TTH, THT, HTT, TTT]

Total number of ways = 2 × 2 × 2 = 8.  Fav. Cases = 7

P (A) = ⅞.

Probability is a very important chapter for your board exams and for the classes to come. It is taught in the 12th standard, and the questions asked; hold considerable marks. If you find it difficult to understand the dice or coin problems or a topic, you can simply download the MSVgo app. There are more than 6000 tutorial videos, quizzes, and solved questions to make learning fun.

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
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  • Carbon And Its Compounds
  • Carboxylic acids and Acid Derivatives
  • Chemical Bonding and Molecular Structures
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  • Chemistry in Everyday Life
  • Chemistry of p-Block elements
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  • Classification of Elements
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  • Cyanide, Isocyanide, Nitro compounds and Amines
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  • Elements, Compounds and Mixtures
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  • Equilibrium
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  • Haloalkanes and Haloarenes
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  • Ideal solutions
  • Introduction to Organic Chemistry
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  • Relative Molecular Mass and Mole
  • States of Matter
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  • Study of Compounds
  • Study of Gas Laws
  • Study of Representative Elements
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  • The d-block and f-block elements
  • The Gaseous State
  • The p-Block Elements
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  • 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
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  • 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
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  • Heredity and Variation
  • How Do Organisms Reproduce?
  • Human Diseases
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  • Life Processes
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  • Molecular Basis of Inheritance
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  • 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
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  • 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
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  • Geometry – Constructions
  • Geometry – Introduction to Euclid’s Geometry
  • Geometry – Three-dimensional Geometry
  • Geometry – Lines and Angles
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  • Geometry – Straight Lines
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  • Linear Programming
  • Matrices and Determinants
  • Mensuration – Areas
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  • 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
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  • 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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