Experimental probability, also known as empirical probability, is a concept in mathematics that deals with estimating the likelihood of an event occurring based on actual experimental results. Unlike theoretical probability, which predicts outcomes based on known possibilities, experimental probability is derived from real-life experiments and observations.
To understand this better, imagine flipping a coin. The theoretical probability of landing heads is 50% or 1/2. However, if you actually flip the coin 100 times and record the outcomes, you might get heads 48 times. The experimental probability of getting heads would then be 48/100 or 0.48.
In this article, we will explore the concept of experimental probability, its significance, and how it differs from theoretical probability. We will discuss the formula for calculating experimental probability, provide examples to illustrate its application.
What is Probability?
The branch of mathematics that tells us about the likelihood of the occurrence of any event is the probability. Probability tells us about the chances of happening an event.
The probability of any element that is sure to occur is One(1) whereas the probability of any impossible event is Zero(0). The probability of all the elements ranges between 0 to 1.
There are two ways of studying probability that are
- Experimental Probability
- Theoretical Probability
Now let's learn about both in detail.
What is Experimental Probability?
Experimental probability is a type of probability that is calculated by conducting an actual experiment or by performing a series of trials to observe the occurrence of an event. It is also known as empirical probability.
To calculate experimental probability, you need to conduct an experiment by repeating the event multiple times and observing the outcomes. Then, you can find the probability of the event occurring by dividing the number of times the event occurred by the total number of trials.
The experimental Probability for Event A can be calculated as follows:
P(E) = (Number of times an event occur in an experiment) / (Total number of Trials)
Examples of Experimental Probability
Now, as we learn the formula, let's put this formula in our coin-tossing case. If we tossed a coin 10 times and recorded a head 4 times and a tail 6 times then the Probability of Occurrence of Head on tossing a coin:
P(H) = 4/10
Similarly, the Probability of Occurrence of Tails on tossing a coin:
P(T) = 6/10
What is Theoretical Probability?
Theoretical Probability deals with assumptions in order to avoid unfeasible or expensive repetition experiments. The theoretical Probability for an Event A can be calculated as follows:
P(A) = Number of outcomes favorable to Event A / Number of all possible outcomes
Now, as we learn the formula, let's put this formula in our coin-tossing case. In tossing a coin, there are two outcomes: Head or Tail.
Hence, The Probability of occurrence of Head on tossing a coin is
P(H) = 1/2
Similarly, The Probability of the occurrence of a Tail on tossing a coin is
P(T) = 1/2
Experimental Probability vs Theoretical Probability
There are some key differences between Experimental and Theoretical Probability, some of which are as follows:
Aspect of Difference | Experimental Probability | Theoretical Probability |
|---|
| Definition | Empirical probability obtained by conducting
experiments or observations | Probability obtained by using mathematical
principles and formulas |
|---|
| Basis | Observed outcomes in real-life experiments | Theoretical predictions based on assumptions
and models |
|---|
| Accuracy | Can be highly variable due to small sample
sizes or other factors | More accurate and reliable, assuming the
assumptions and models are correct |
|---|
| Calculation | Calculated by dividing the number of times
an event occurred by the total number of trials | Calculated by dividing the number of favorable
outcomes by the total number of possible outcomes |
|---|
| Application | Used when data is collected through
experimentation or observation | Used when predicting outcomes for theoretical scenarios |
|---|
| Examples | Tossing a coin or rolling a die multiple times
to determine the probability of an event | Calculating the probability of drawing a certain
card from a deck or the probability of winning
a game with specific rules |
|---|
Read More,
Solved Examples of Experimental Probability
Example 1. Let's take an example of tossing a coin, tossing it 40 times, and recording the observations. By using the formula, we can find the experimental probability for heads and tails as shown in the below table.
Answer:
| Number of Trail | Outcome | Number of Trail | Outcome | Number of Trail | Outcome | Number of Trail | Outcome |
|---|
First | H | Eleventh | T | Twenty-first | T | Thirty-first | T |
Second | T | Twelfth | T | Twenty-second | H | Thirty-second | H |
Third | T | Thirteenth | H | Twenty-third | T | Thirty-third | T |
Fourth | H | Fourteenth | H | Twenty-fourth | H | Thirty-fourth | H |
Fifth | H | Fifteenth | H | Twenty-fifth | T | Thirty-fifth | T |
Sixth | H | Sixteenth | H | Twenty-sixth | H | Thirty-sixth | T |
Seventh | T | Seventeenth | T | Twenty-seventh | T | Thirty-seventh | T |
Eighth | H | Eighteenth | T | Twenty-eighth | T | Thirty-eighth | H |
Ninth | T | Nineteenth | T | Twenty-ninth | T | Thirty-ninth | T |
Tenth | H | Twentieth | T | Thirtieth | H | Fortieth | T |
The formula for experimental probability:
P(H) = Number of Heads ÷ Total Number of Trials = 16 ÷ 40 = 0.4
Similarly,
P(H) = Number of Tails ÷ Total Number of Trials = 24 ÷ 40 = 0.6
P(H) + P(T) = 0.6 + 0.4 = 1
Note: Repeat this experiment for 'n' times and then you will find that the number of times increases, the fraction of experimental probability comes closer to 0.5. Thus if we add P(H) and P(T), we will get 0.6 + 0.4 = 1 which means P(H) and P(T) is the only possible outcomes.
Example 2. A manufacturer makes 50,000 cell phones every month. After inspecting 1000 phones, the manufacturer found that 30 phones are defective. What is the probability that you will buy a phone that is defective? Predict how many phones will be defective next month.
Answer:
Experimental Probability = 30/1000 = 0.03
0.03 = (3/100) × 100 = 3%
The probability that you will buy a defective phone is 3%
⇒ Number of defective phones next month = 3% × 50000
⇒ Number of defective phones next month = 0.03 × 50000
⇒ Number of defective phones next month = 1500
Example 3. There are about 320 million people living in the USA. Pretend that a survey of 1 million people revealed that 300,000 people think that all cars should be electric. What is the probability that someone chosen randomly does not like the electric car? How many people like electric cars?
Answer:
Now the number of people who do not like electric cars is 1000000 - 300000 = 700000
Experimental Probability = 700000/1000000 = 0.7
And, 0.7 = (7/10) × 100 = 70%
The probability that someone chose randomly does not like the electric car is 70%
The probability that someone like electric cars is 300000/1000000 = 0.3
Let x be the number of people who love electric cars
⇒ x = 0.3 × 320 million
⇒ x = 96 million
The number of people who love electric cars is 96 million.
Practice Problems on Experimental Probability
Problem 1: A coin is flipped 200 times, and it lands on heads 120 times. What is the experimental probability of getting heads?
Problem 2: A die is rolled 50 times, and the number 3 appears 8 times. What is the experimental probability of rolling a 3?
Problem 3: In a class survey, 150 students were asked if they prefer reading books or watching movies. 90 students said they prefer watching movies. What is the experimental probability that a randomly chosen student prefers watching movies?
Problem 4: A bag contains 5 red, 7 blue, and 8 green marbles. If 40 marbles are drawn at random with replacement, and 12 of them are red, what is the experimental probability of drawing a red marble?
Problem 5: A basketball player made 45 successful free throws out of 60 attempts. What is the experimental probability that the player will make a free throw?
Problem 6: During a game, a spinner is spun 80 times, landing on a specific section 20 times. What is the experimental probability of the spinner landing on that section?
Similar Reads
CBSE Class 10 Maths Notes PDF: Chapter Wise Notes 2024 Math is an important subject in CBSE Class 10th Board exam. So students are advised to prepare accordingly to score well in Mathematics. Mathematics sometimes seems complex but at the same time, It is easy to score well in Math. So, We have curated the complete CBSE Class 10 Math Notes for you to pr
15+ min read
Chapter 1: Real Numbers
Real NumbersReal Numbers are continuous quantities that can represent a distance along a line, as Real numbers include both rational and irrational numbers. Rational numbers occupy the points at some finite distance and irrational numbers fill the gap between them, making them together to complete the real line
10 min read
Euclid Division LemmaEuclid's Division Lemma which is one of the fundamental theorems proposed by the ancient Greek mathematician Euclid which was used to prove various properties of integers. The Euclid's Division Lemma also serves as a base for Euclid's Division Algorithm which is used to find the GCD of any two numbe
5 min read
Fundamental Theorem of ArithmeticThe Fundamental Theorem of Arithmetic is a useful method to understand the prime factorization of any number. The factorization of any composite number can be uniquely written as a multiplication of prime numbers, regardless of the order in which the prime factors appear. The figures above represent
6 min read
HCF / GCD and LCM - Definition, Formula, Full Form, ExamplesThe full form of HCF/GCD is the Highest Common Factor/Greatest Common Divisor(Both terms mean the same thing), while the full form of LCM is the Least Common Multiple. HCF is the largest number that divides two or more numbers without leaving a remainder, whereas LCM is the smallest multiple that is
9 min read
Irrational Numbers- Definition, Examples, Symbol, PropertiesIrrational numbers are real numbers that cannot be expressed as fractions. Irrational Numbers can not be expressed in the form of p/q, where p and q are integers and q ≠0.They are non-recurring, non-terminating, and non-repeating decimals. Irrational numbers are real numbers but are different from
12 min read
Decimal Expansions of Rational NumbersReal numbers are simply the combination of rational and irrational numbers, in the number system. In general, all the arithmetic operations can be performed on these numbers and they can be represented in the number line, also. So in this article let's discuss some rational and irrational numbers an
6 min read
Rational NumbersRational numbers are a fundamental concept in mathematics, defined as numbers that can be expressed as the ratio of two integers, where the denominator is not zero. Represented in the form p/q​ (with p and q being integers), rational numbers include fractions, whole numbers, and terminating or repea
15+ min read
Chapter 2: Polynomials
Algebraic Expressions in Math: Definition, Example and EquationAlgebraic Expression is a mathematical expression that is made of numbers, and variables connected with any arithmetical operation between them. Algebraic forms are used to define unknown conditions in real life or situations that include unknown variables.An algebraic expression is made up of terms
8 min read
Polynomial FormulaThe polynomial Formula gives the standard form of polynomial expressions. It specifies the arrangement of algebraic expressions according to their increasing or decreasing power of variables. The General Formula of a Polynomial:f(x) = an​xn + an−1​xn−1 + ⋯ + a1​x + a0​Where,an​, an−1​, …, a1​, a0​ a
5 min read
Types of Polynomials (Based on Terms and Degrees)Types of Polynomials: In mathematics, an algebraic expression is an expression built up from integer constants, variables, and algebraic operations. There are mainly four types of polynomials based on degree-constant polynomial (zero degree), linear polynomial ( 1st degree), quadratic polynomial (2n
9 min read
Zeros of PolynomialZeros of a Polynomial are those real, imaginary, or complex values when put in the polynomial instead of a variable, the result becomes zero (as the name suggests zero as well). Polynomials are used to model some physical phenomena happening in real life, they are very useful in describing situation
13 min read
Geometrical meaning of the Zeroes of a PolynomialAn algebraic identity is an equality that holds for any value of its variables. They are generally used in the factorization of polynomials or simplification of algebraic calculations. A polynomial is just a bunch of algebraic terms added together, for example, p(x) = 4x + 1 is a degree-1 polynomial
8 min read
Factorization of PolynomialFactorization in mathematics refers to the process of expressing a number or an algebraic expression as a product of simpler factors. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, and we can express 12 as 12 = 1 × 12, 2 × 6, or 4 × 3.Similarly, factorization of polynomials involves expre
10 min read
Division Algorithm for PolynomialsPolynomials are those algebraic expressions that contain variables, coefficients, and constants. For Instance, in the polynomial 8x2 + 3z - 7, in this polynomial, 8,3 are the coefficients, x and z are the variables, and 7 is the constant. Just as simple Mathematical operations are applied on numbers
5 min read
Algebraic IdentitiesAlgebraic Identities are fundamental equations in algebra where the left-hand side of the equation is always equal to the right-hand side, regardless of the values of the variables involved. These identities play a crucial role in simplifying algebraic computations and are essential for solving vari
14 min read
Relationship between Zeroes and Coefficients of a PolynomialPolynomials are algebraic expressions with constants and variables that can be linear i.e. the highest power o the variable is one, quadratic and others. The zeros of the polynomials are the values of the variable (say x) that on substituting in the polynomial give the answer as zero. While the coef
9 min read
Division Algorithm Problems and SolutionsPolynomials are made up of algebraic expressions with different degrees. Degree-one polynomials are called linear polynomials, degree-two are called quadratic and degree-three are called cubic polynomials. Zeros of these polynomials are the points where these polynomials become zero. Sometimes it ha
6 min read
Chapter 3: Pair of Linear equations in two variables
Linear Equation in Two VariablesLinear Equation in Two Variables: A Linear equation is defined as an equation with the maximum degree of one only, for example, ax = b can be referred to as a linear equation, and when a Linear equation in two variables comes into the picture, it means that the entire equation has 2 variables presen
9 min read
Pair of Linear Equations in Two VariablesLinear Equation in two variables are equations with only two variables and the exponent of the variable is 1. This system of equations can have a unique solution, no solution, or an infinite solution according to the given initial condition. Linear equations are used to describe a relationship betwe
11 min read
Graphical Methods of Solving Pair of Linear Equations in Two VariablesA system of linear equations is just a pair of two lines that may or may not intersect. The graph of a linear equation is a line. There are various methods that can be used to solve two linear equations, for example, Substitution Method, Elimination Method, etc. An easy-to-understand and beginner-fr
8 min read
Solve the Linear Equation using Substitution MethodA linear equation is an equation where the highest power of the variable is always 1. Its graph is always a straight line. A linear equation in one variable has only one unknown with a degree of 1, such as:3x + 4 = 02y = 8m + n = 54a – 3b + c = 7x/2 = 8There are mainly two methods for solving simult
10 min read
Solving Linear Equations Using the Elimination MethodIf an equation is written in the form ax + by + c = 0, where a, b, and c are real integers and the coefficients of x and y, i.e. a and b, are not equal to zero, it is said to be a linear equation in two variables. For example, 3x + y = 4 is a linear equation in two variables- x and y. The numbers th
9 min read
Chapter 4: Quadratic Equations
Quadratic EquationsA Quadratic equation is a second-degree polynomial equation that can be represented as ax2 + bx + c = 0. In this equation, x is an unknown variable, a, b, and c are constants, and a is not equal to 0. The solutions of a quadratic equation are known as its roots. These roots can be found using method
12 min read
Roots of Quadratic EquationThe roots of a quadratic equation are the values of x that satisfy the equation. The roots of a quadratic equation are also called zeros of a quadratic equation. A quadratic equation is generally in the form: ax2 + bx + c = 0Where:a, b, and c are constants (with a ≠0).x represents the variable.Root
13 min read
Solving Quadratic EquationsA quadratic equation, typically in the form ax² + bx + c = 0, can be solved using different methods including factoring, completing the square, quadratic formula, and the graph method. While Solving Quadratic Equations we try to find a solution that represent the points where this the condition Q(x)
8 min read
How to find the Discriminant of a Quadratic Equation?Algebra can be defined as the branch of mathematics that deals with the study, alteration, and analysis of various mathematical symbols. It is the study of unknown quantities, which are often depicted with the help of variables in mathematics. Algebra has a plethora of formulas and identities for th
4 min read
Chapter 5: Arithmetic Progressions
Arithmetic Progressions Class 10- NCERT NotesArithmetic Progressions (AP) are fundamental sequences in mathematics where each term after the first is obtained by adding a constant difference to the previous term. Understanding APs is crucial for solving problems related to sequences and series in Class 10 Mathematics. These notes cover the ess
7 min read
Sequences and SeriesA sequence is an ordered list of numbers following a specific rule. Each number in a sequence is called a "term." The order in which terms are arranged is crucial, as each term has a specific position, often denoted as an​, where n indicates the position in the sequence.For example:2, 5, 8, 11, 14,
10 min read
Arithmetic Progression in MathsArithmetic Progression (AP) or Arithmetic Sequence is simply a sequence of numbers such that the difference between any two consecutive terms is constant.Some Real World Examples of APNatural Numbers: 1, 2, 3, 4, 5, . . . with a common difference 1Even Numbers: 2, 4, 6, 8, 10, . . . with a common di
3 min read
Arithmetic Progression - Common difference and Nth term | Class 10 MathsArithmetic Progression is a sequence of numbers where the difference between any two successive numbers is constant. For example 1, 3, 5, 7, 9....... is in a series which has a common difference (3 - 1) between two successive terms is equal to 2. If we take natural numbers as an example of series 1,
5 min read
How to find the nth term of an Arithmetic Sequence?Answer - Use the formula: an = a1 + (n - 1)dWhere:an = nth term,a = first term,d = common difference,n = term number.Substitute the values of a, d, and n into the formula to calculate an.Steps to find the nth Term of an Arithmetic SequenceStep 1: Identify the First and Second Term: 1st and 2nd term,
3 min read
Arithmetic Progression – Sum of First n Terms | Class 10 MathsArithmetic Progression is a sequence of numbers where the difference between any two successive numbers is constant. For example 1, 3, 5, 7, 9……. is in a series which has a common difference (3 – 1) between two successive terms is equal to 2. If we take natural numbers as an example of series 1, 2,
8 min read
Arithmetic MeanArithmetic Mean, commonly known as the average, is a fundamental measure of central tendency in statistics. It is defined as the ratio of all the values or observations to the total number of values or observations. Arithmetic Mean is one of the fundamental formulas used in mathematics and it is hig
12 min read
Arithmetic Progression – Sum of First n Terms | Class 10 MathsArithmetic Progression is a sequence of numbers where the difference between any two successive numbers is constant. For example 1, 3, 5, 7, 9……. is in a series which has a common difference (3 – 1) between two successive terms is equal to 2. If we take natural numbers as an example of series 1, 2,
8 min read
Chapter 6: Triangles
Triangles in GeometryA triangle is a polygon with three sides (edges), three vertices (corners), and three angles. It is the simplest polygon in geometry, and the sum of its interior angles is always 180°. A triangle is formed by three line segments (edges) that intersect at three vertices, creating a two-dimensional re
13 min read
Similar TrianglesSimilar Triangles are triangles with the same shape but can have variable sizes. Similar triangles have corresponding sides in proportion to each other and corresponding angles equal to each other. Similar triangles are different from congruent triangles. Two congruent figures are always similar, bu
15 min read
Criteria for Similarity of TrianglesThings are often referred similar when the physical structure or patterns they show have similar properties, Sometimes two objects may vary in size but because of their physical similarities, they are called similar objects. For example, a bigger Square will always be similar to a smaller square. In
9 min read
Basic Proportionality Theorem (BPT) Class 10 | Proof and ExamplesBasic Proportionality Theorem: Thales theorem is one of the most fundamental theorems in geometry that relates the parts of the length of sides of triangles. The other name of the Thales theorem is the Basic Proportionality Theorem or BPT. BPT states that if a line is parallel to a side of a triangl
8 min read
Pythagoras Theorem | Formula, Proof and ExamplesPythagoras Theorem explains the relationship between the three sides of a right-angled triangle and helps us find the length of a missing side if the other two sides are known. It is also known as the Pythagorean theorem. It states that in a right-angled triangle, the square of the hypotenuse is equ
9 min read
Chapter 7: Coordinate Geometry
Coordinate GeometryCoordinate geometry is a branch of mathematics that combines algebra and geometry using a coordinate plane. It helps us represent points, lines, and shapes with numbers and equations, making it easier to analyze their positions, distances, and relationships. From plotting points to finding the short
3 min read
Distance formula - Coordinate Geometry | Class 10 MathsThe distance formula is one of the important concepts in coordinate geometry which is used widely. By using the distance formula we can find the shortest distance i.e drawing a straight line between points. There are two ways to find the distance between points:Pythagorean theoremDistance formulaTab
9 min read
Distance Between Two PointsDistance Between Two Points is the length of line segment that connects any two points in a coordinate plane in coordinate geometry. It can be calculated using a distance formula for 2D or 3D. It represents the shortest path between two locations in a given space.In this article, we will learn how t
6 min read
Section FormulaSection Formula is a useful tool in coordinate geometry, which helps us find the coordinate of any point on a line which is dividing the line into some known ratio. Suppose a point divides a line segment into two parts which may be equal or not, with the help of the section formula we can find the c
14 min read
How to find the ratio in which a point divides a line?Answer: To find the ratio in which a point divides a line we use the following formula x = \frac{m_1x_2+m_2x_1}{m_1+m_2}Â Â y = \frac{m_1y_2+m_2y_1}{m_1+m_2}Geo means Earth and metry means measurement. Geometry is a branch of mathematics that deals with distance, shapes, sizes, relevant positions of a
4 min read
How to find the Trisection Points of a Line?To find the trisection points of a line segment, you need to divide the segment into three equal parts. This involves finding the points that divide the segment into three equal lengths. In this article, we will answer "How to find the Trisection Points of a Line?" in detail including section formul
4 min read
How to find the Centroid of a Triangle?Answer: The Centroid for the triangle is calculated using the formula\left (\frac{[x1+x2+x3]}{3}, \frac{[y1+y2+y3]}{3}\right)A triangle consists of three sides and three interior angles. Centroid refers to the center of an object. Coming to the centroid of the triangle, is defined as the meeting poi
4 min read
Area of a Triangle in Coordinate GeometryThere are various methods to find the area of the triangle according to the parameters given, like the base and height of the triangle, coordinates of vertices, length of sides, etc. In this article, we will discuss the method of finding area of any triangle when its coordinates are given.Area of Tr
6 min read
Chapter 8: Introduction to Trigonometry
Trigonometric RatiosThere are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratio. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).As give
4 min read
Unit Circle: Definition, Formula, Diagram and Solved ExamplesUnit Circle is a Circle whose radius is 1. The center of unit circle is at origin(0,0) on the axis. The circumference of Unit Circle is 2π units, whereas area of Unit Circle is π units2. It carries all the properties of Circle. Unit Circle has the equation x2 + y2 = 1. This Unit Circle helps in defi
7 min read
Trigonometric Ratios of Some Specific AnglesTrigonometry is all about triangles or to be more precise the relationship between the angles and sides of a triangle (right-angled triangle). In this article, we will be discussing the ratio of sides of a right-angled triangle concerning its acute angle called trigonometric ratios of the angle and
6 min read
Trigonometric IdentitiesTrigonometric identities play an important role in simplifying expressions and solving equations involving trigonometric functions. These identities, which include relationships between angles and sides of triangles, are widely used in fields like geometry, engineering, and physics. Some important t
10 min read
Chapter 9: Some Applications of Trigonometry