Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Data Handling → Probability ↓


Experimental Probability


Probability is a branch of mathematics that deals with the likelihood of an event occurring. It answers questions such as "What are the chances of winning the game?" or "How likely is it to rain tomorrow?" In mathematics, probability is expressed as a number between 0 and 1. A probability of 0 means that the event will not happen, while a probability of 1 means that the event will definitely happen.

What is experimental probability?

Experimental probability is a way of determining the probability of an event. Instead of calculating what should happen in theory, we look at what actually happens. This type of probability is based on actual experiments or trials. It is determined by performing an event repeatedly and collecting data from these experiments. Through this process, we can estimate the probability of different outcomes.

The formula for experimental probability is simple:

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

This formula calculates the proportion of trials in which a particular event occurs. As the number of trials increases, the experimental probability approaches the theoretical probability.

Doing an experiment

To better understand experimental probability, let's consider a simple experiment. Suppose you have a six-sided dice, and you want to find the experimental probability of getting a 4.

  1. Roll the dice 100 times.
  2. Record how many times you threw a 4.

Suppose that after rolling 100 times, you rolled a 4 a total of 18 times. Then the experimental probability of rolling a 4 is calculated as follows:

Experimental Probability = 18/100 = 0.18

This result means that based on your experiment, the probability of getting a 4 is 0.18. This experimental probability is not exactly 1/6 or approximately 0.167, which is the theoretical probability because the experiment was only performed 100 times, not an infinite number of times.

Visualization example: coin toss

A common example to understand experimental probability is tossing a coin. A fair coin has two sides, heads (H) and tails (T). The theoretical probability of getting heads is 0.5. Let us perform an experiment to find the experimental probability of getting heads.

Suppose we toss a coin 50 times and record the results:

H Tea H Tea H

This chart shows the results of tossing a coin 50 times. The height of each bar shows how many times heads or tails came up.

Suppose we got heads 28 times. Using the formula for experimental probability:

Experimental Probability of Heads = 28/50 = 0.56

Thus, according to our experiment, the probability of getting heads is 0.56, which is slightly different from the theoretical probability of 0.50.

Large number of trials and the law of large numbers

As you increase the number of trials, the experimental probability should approach the theoretical probability. This phenomenon is explained by the "law of large numbers," which states that as the number of trials in an experiment increases, the experimental probability of an event will approach its theoretical probability.

For example, if you flip a coin 1,000 times, the number of heads divided by 1,000 will be very close to 0.5. With more attempts, the effects of fluctuations in probability are reduced, leading to more accurate estimates of probability.

Another visual example: spinner

Let's consider another example in which a spinner is divided into three equal parts: red, blue and green. The spinner is fair, so the theoretical probability of landing on any color is:

1/3 ≈ 0.333

Suppose you spin the spinner 60 times and record the results. If you land on red 20 times, blue 15 times, and green 25 times, calculate the experimental probabilities.

The experimental probabilities are calculated as follows:

  • Red: 20/60 = 0.333
  • Blue: 15/60 = 0.25
  • Green: 25/60 = 0.416

In this experiment, we see variations due to chance. The experimental probability for red closely matches the theoretical value, while blue and green differ, highlighting the role of chance in small sample sizes.

Using experimental probability to make decisions

Experimental probability is not only a tool for understanding probability but also helps in decision-making processes. For example, scientists often use experimental probability to predict outcomes and test hypotheses in real-world scenarios. Engineers can use it to assess the reliability of machines by recording the number of failures across multiple tests.

As an everyday example, consider a person buying a lottery ticket. By experimenting – buying multiple tickets over time – they can record how often they win and inform their future purchasing decisions based on the experimental probability of winning.

Limitations of experimental probability

While experimental probability is valuable, it has limitations. Results can vary considerably with a small number of trials due to chance. Suppose that in some throws of dice, you throw more sixes than theoretically expected. This does not change the theoretical probability, but highlights the importance of sample size.

In addition, errors can arise from unfair dice or biased methods. Accurate representation of probability depends on fair and unbiased conditions.

Conclusion

Understanding experimental probability leads to an understanding of how probability is assessed in practice. Through real-world experiments, it becomes clear that probability is not always exact but rather an estimate. Alignment between experimental and theoretical probabilities may require extensive trials, demonstrating the essence of the law of large numbers. Despite potential limitations, experimental probability is a foundational tool within mathematics and practical applications, informing decisions and illuminating patterns within random events.


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Experimental Probability

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