Grade 10
  1. Number Systems  1.1. Real Numbers  1.1.1. Properties of Real Numbers  1.1.2. Rational and Irrational Numbers  1.1.3. Decimal Representation of Real Numbers  1.1.4. Operations on Real Numbers  1.1.5. Real Numbers on the Number Line  1.2. Euclid's Division Lemma  1.3. Prime Factorization  1.4. LCM and HCF  1.5. Exponents and Radicals  1.5.1. Laws of Exponents  1.5.2. Simplifying Radical Expressions  1.5.3. Scientific Notation
  2. Algebra  2.1. Polynomials  2.1.1. Definition and Types of Polynomials  2.1.2. Degree of a Polynomial  2.1.3. Zeros of a Polynomial  2.1.4. Factorization of Polynomials  2.1.5. Algebraic Identities  2.2. Linear Equations in Two Variables  2.2.1. Graph of Linear Equations  2.2.2. Solutions of Linear Equations  2.2.3. Application of Linear Equations in Word Problems  2.3. Quadratic Equations  2.3.1. Standard Form of a Quadratic Equation  2.3.2. Methods of Solving Quadratic Equations  2.3.2.1. Factorization Method  2.3.2.2. Completing the Square Method  2.3.2.3. Quadratic Formula  2.3.3. Nature of Roots  2.4. Arithmetic Progressions  2.4.1. Definition of Arithmetic Progression  2.4.2. General Term of an AP  2.4.3. Sum of First n Terms of an AP  2.5. Introduction to Functions  2.5.1. Definition and Types of Functions  2.5.2. Domain and Range  2.5.3. Composition of Functions  2.5.4. Inverse of a Function
  3. Coordinate Geometry  3.1. Cartesian System  3.2. Distance Formula  3.3. Section Formula  3.4. Area of a Triangle  3.5. Slope of a Line  3.6. Equation of a Line  3.6.1. Slope-Intercept Form  3.6.2. Point-Slope Form  3.6.3. Two-Point Form  3.6.4. Intercept Form  3.6.5. General Form of a Line
  4. Trigonometry  4.1. Trigonometric Ratios  4.1.1. Sine Cosine Tangent and their Reciprocals  4.1.2. Values of Trigonometric Ratios at Specific Angles  4.2. Trigonometric Identities  4.3. Trigonometric Ratios of Complementary Angles  4.4. Heights and Distances  4.4.1. Angle of Elevation and Depression  4.4.2. Applications in Real-Life Problems
  5. Geometry  5.1. Triangles  5.1.1. Congruence of Triangles  5.1.2. Criteria for Congruence  5.1.3. Properties of Triangles  5.2. Similarity  5.2.1. Similar Figures  5.2.2. Criteria for Similarity  5.2.3. Similarity of Triangles  5.2.4. Applications of Similarity in Real-Life  5.3. Circles  5.3.1. Properties of a Circle  5.3.2. Tangent to a Circle  5.3.3. Number of Tangents from a Point  5.3.4. Angle Subtended by an Arc  5.4. Constructions  5.4.1. Construction of Similar Triangles  5.4.2. Tangent to a Circle  5.4.3. Construction of Angle Bisectors
  6. Mensuration  6.1. Areas of Plane Figures  6.1.1. Area of a Triangle  6.1.2. Area of a Parallelogram  6.1.3. Area of a Trapezium  6.1.4. Area of a Rhombus  6.2. Surface Areas and Volumes  6.2.1. Surface Area of a Cube Cuboid and Cylinder  6.2.2. Surface Area of a Cone and Sphere  6.2.3. Volume of a Cube Cuboid and Cylinder  6.2.4. Volume of a Cone and Sphere  6.3. Conversion of Units  6.4. Frustum of a Cone
  7. Statistics  7.1. Introduction to Statistics  7.2. Collection of Data  7.3. Presentation of Data  7.3.1. Tabular Form  7.3.2. Graphical Form  7.3.2.1. Bar Graph  7.3.2.2. Histogram  7.3.2.3. Frequency Polygon  7.3.2.4. Ogive  7.4. Measures of Central Tendency  7.4.1. Mean Median and Mode  7.4.2. Quartiles and Percentiles  7.5. Measures of Dispersion  7.5.1. Range and Interquartile Range  7.5.2. Standard Deviation and Variance
  8. Probability  8.1. Introduction to Probability  8.2. Experimental Probability  8.3. Theoretical Probability  8.4. Probability of Simple Events  8.5. Complementary Events  8.6. Probability of Compound Events

Grade 10 → Probability ↓


Experimental Probability


Probability is a fascinating field in mathematics that attempts to estimate how likely events are. It finds use in many fields such as gambling, statistics, finance, science, and much more. In grade 10 math, an important concept related to probability that we learn is “experimental probability.” This concept is important for understanding the difference between theoretical probability and what happens in real life when we perform experiments or tests.

What is experimental probability?

Experimental probability is the probability calculated based on the results of an actual experiment. Unlike theoretical probability, which is based on how events are expected to occur in theory, experimental probability allows us to look at what actually happens when we perform a test.

Mathematically, experimental probability is defined as:

Experimental Probability = (Number of times an event occurs) / (Total number of trials)

This means: - "Number of occurrences of the event" means how many times you get the outcome you are interested in. - "Total number of attempts" means how many times you have performed the experiment or activity.

A simple example

Imagine that you roll a six-sided die 100 times. You want to find the experimental probability of getting a 4. You roll the die and observe that 4 comes up 18 times.

Use the formula we just saw:

Experimental Probability of rolling a 4 = 18 / 100 = 0.18

In this example, the experimental probability of getting a 4 is 0.18, which means the probability of getting a 4 based on your experiment is 18%.

Visualization example: coin toss

Another classic example of experimental probability is coin tossing. Let’s try to understand it with the help of visualization.

Head Tail

Suppose you flip a coin 50 times. In real scenarios, average people expect "heads" to come up about half the time and "tails" half the time. But this may not always be the case. Suppose the outcome of your experiment is 28 heads and 22 tails.

To find the experimental probability of getting heads, you apply the formula:

Experimental Probability of getting heads = 28 / 50 = 0.56

Similarly, to find the experimental probability of tails:

Experimental Probability of getting tails = 22 / 50 = 0.44

This means that the experimental probability based on your experiment is 56% for heads and 44% for tails.

Relating experimental probability to theoretical probability

In theoretical probability, you would expect a fair coin to have a probability of 0.5 for heads and 0.5 for tails because both outcomes are equally likely. However, the experimental probability sometimes differs from the theoretical probability due to chance and the number of trials.

The main idea is that as you increase the number of trials (or experiments), the experimental probability approaches the theoretical probability. This is known as the "law of large numbers."

Example with dice

Let us consider throwing a six-sided dice. The theoretical probability of getting any number (say, 3) is

Probability (3) = 1/6 ≈ 0.166

If you roll a dice 100 times in an experiment and get a 3 only 15 times, then the experimental probability is:

Experimental Probability (3) = 15 / 100 = 0.15

Here, the experimental and theoretical probabilities are close. If you continue to throw the dice and increase the number of trials to 1000, you may find that the experimental probability is getting even closer to 0.166.

Understanding through more visual examples

Dice roll visualization

Let's imagine that two six-sided dice are being tossed simultaneously, and you are interested in the probability of the sum being 7. Below is a simplified illustration of the possible totals when tossing two dice:

2 3 4 7 8 9 12

Suppose in this experiment you throw two dice 200 times, and 30 times you find that the sum is equal to 7.

Experimental Probability (sum of 7) = 30 / 200 = 0.15

Therefore, in this experiment, the probability of getting the sum 7 is 0.15 (which is 15%).

Factors affecting experimental probability

  • Actual results obtained from the experiment.
  • The number of tests or experiments conducted.
  • Random variables affecting experimental results.
  • Precision of measurement and observation techniques.

The more tests you perform, the more reliable the experimental probability. This concept highlights the importance of performing the appropriate number of experiments to reach a reliable result.

Conclusion

In short, experimental probability helps us understand how often an event will occur when tested in the real world. It provides a practical perspective on theoretical concepts by dealing with the actual results of conducted tests and experiments. As you tinker with it and perform more experiments, the difference between expected and observed results decreases, giving you deeper insights into the ever-fascinating world of probability.


Grade 10 → 8.2


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Introduction to Probability
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Experimental Probability

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