# Chapter 11 – Probability

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

Introduction

It is a measurement of the likelihood that a particular event will occur. It ranges between 0 to 1, where 0 means an event is impossible to occur and 1 means an event will occur for sure (certain).

But wait!! Why are we going to study probability? Why should we bother about the likelihood of any event? Is there any practical implication in real life?

Probability definition in maths for class 10th is simply defined as the event divided by the total number of outcomes. You can also use a Probability tree to find the sample space and solve questions easily.

#### The formula for Probability

P= (number of favourable outcomes/ total number of outcomes)= E/S

Where,

• E is the number of favorable outcomes.
• S is the sample space.

Example 1: Suppose you toss a coin; the possible outcomes would be Head or Tail. So, in this case, the exhaustive cases are 2, namely {Head, Tail}. Now, calculate the chances of getting ahead as

Favorable cases/Exhaustive cases = 1/2 = 0.5 is the Probability

Note that the favorable case(s) of getting a head is only one.

Example 2: Suppose you throw a dice, then the possible outcomes, namely, the exhaustive cases, are 6, {1,2,3,4,5,6}. If you consider getting an odd number as a success, let us find out the favorable cases.

The favorable cases are {1,3,5}

Now, you can get the probability of an event of getting an odd number as

Favorable cases/Exhaustive cases = 3/6 = 0.5

So, the required probability is 0.5.

Example 3: Suppose you draw a card from the well-shuffled pack of 52 cards. Let us say that you are interested in finding out the probability that you get a Jack.

Then as a first step, find out the favorable cases:

The favorable cases are 4, namely Jack of Heart, Spade, Clubs, and Diamond.

The Exhaustive cases = 52 cards

Favorable cases/Exhaustive cases = 4/52 = 0.077

So, the required Probability is 0.077

#### Types of Probability

The three primary types of probabilities in practice can be viewed as:

• Empirical – based on data, e.g., the probability is left-handed. The best way to estimate this is to survey a group of people and determine the percent of lefties.
• Theoretical Probability: Based on assumptions and calculations, e.g., the probability of getting a full house on a five-card deal. Here we assume each set of five cards from the pack is equally likely; from this, we can calculate the probability without actually dealing with any cards.
• Subjective – Based on intuition, e.g., what are the Probability aliens will visit Earth by 2100. Here there is no empirical evidence or reasonable assumptions to be made.
• Experimental Probability: When an event is repeated multiple times, the probability taken out from that event is experimental probability.

#### Complementary Events

Complementary Probability is any Probability when one event occurs only when the other complementary event does not occur. 1 is the sum of the two events of a complementary Probability. The complementary event is the exact opposite of each other. For example, take going to school and not going to school. In this case, both events are opposite to each other, and their Probability value sums up to 1.

Applications of Probability:

• Insurance
• Gambling
• Financial mathematics (especially to model options)
• Design (engineering)
• Clinical trials
• Statistics, especially significance in statistical tests
• Physics: statistical physics
• Quantum mechanics
• Queue theorem: Designing telecom networks
• Economics: Extensively used to calculate various rates based on the available metrics and data.

#### Conclusion

Probability lets us know how likely an event is to occur where all the sample space events have an equal chance. There are different types of probability and have been discussed above. Probability is a very widely used concept that has its applications in most industries and education. Suppose you want to pursue a career in probability and statistics. In that case, you can opt for an undergraduate degree in statistics from reputed colleges like ISI in India.

#### FAQs

1. How do you calculate probability?

Probability tells us the likeliness of any event from a given set of all probable events. We calculate it by a formula.

P= (Desired result/ Total number of outcomes)

1. What are the 5 rules of probability?
• Rule 1: For any possible event A, 0 ≤ P(A) ≤ 1.
• Rule 2: The total sum of probabilities of all the possible outcomes is 1.
• Rule 3: P(not A) = (1 – P(A))
o    the probability that the event does not happen is 1 minus the probability that it does happen.
• Rule 4: If A & B are disjoint events, then P(A or B) = P(A) + P(B).
• Rule 5: The General Addition Rule: P(A or B) = P(A) + P(B) – P(A and B)
1.     What are the four types of probability?
•         Classical P(A)=m/n
•         Empirical P(A)= the limit as n approaches Infinity of m/n
•         Subjective: It is an individual measure of belief that an event is likely to occur.
•         Axiomatic Probability: The coherence condition is needed to validate the subjective probability with classical and empirical probability.
1.     How do you find a Probability example?

You can find Probability examples on MSVgo, where each example is based on different types of Probability questions. A simple example is given below.

Example 1: Suppose you toss a coin; the possible outcomes would be Head or Tail. So, in this case, the exhaustive cases are 2, namely {Head, Tail}. Now, calculate the chances of getting a head as

Favorable cases/Exhaustive cases = 1/2 = 0.5 is the Probability

Note that the favorable case(s) of getting a head is only one.

1.     Should you simplify probability?

Yes, you can simplify probability to get results easily. Divide the numerator & the denominator by the highest common factor.

On MSVgo, you can learn the meaning of probability intuitively by examples and animations.

### 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
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• Elements, Compounds and Mixtures
• Environmental Chemistry
• Equilibrium
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• Haloalkanes and Haloarenes
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• Matter
• Matter Around Us
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• Metallurgy
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• Mole Concept and Stoichiometry
• Natural Resources
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• Periodic Classification of Elements
• Physical and Chemical Changes
• Physical and Chemical Properties of Water
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• 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
• 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

• 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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