Probability - Experimental Approach
Probability is the branch of mathematics that measures the likelihood of an event occurring. It assigns a numerical value between 0 and 1 to every event, where 0 means impossible and 1 means certain.
In Class 9 Mathematics (NCERT Chapter 15: Probability), the focus is on the experimental (empirical) approach to probability. Unlike the theoretical approach (studied in Class 10), experimental probability is calculated by actually performing an experiment many times, recording the outcomes, and computing the ratio of favourable outcomes to total trials.
The word "probability" comes from the Latin word probabilis, meaning "worthy of approval" or "likely". The mathematical study of probability began in the 17th century with the work of Blaise Pascal and Pierre de Fermat, who analysed games of chance.
Probability is used extensively in weather forecasting ("30% chance of rain"), medicine ("95% effectiveness"), insurance (risk assessment), sports (win predictions), and quality control (defect rates). Understanding probability helps us make better decisions under uncertainty.
The key difference between experimental and theoretical probability is that experimental probability is based on what actually happened in an experiment, while theoretical probability is based on what should happen in an ideal scenario. As the number of trials increases, experimental probability converges to theoretical probability — this is the Law of Large Numbers.
What is Probability - Experimental Approach?
Definition: The experimental probability (or empirical probability) of an event E is defined as:
P(E) = Number of trials in which event E occurred / Total number of trials
Where:
- P(E) = probability of event E (a number between 0 and 1)
- Numerator = number of trials where the favourable outcome was observed
- Denominator = total number of times the experiment was performed
Key terminology:
- Experiment (Random Experiment): An action whose outcome is uncertain. Examples: tossing a coin, rolling a die, drawing a card from a deck, picking a marble from a bag.
- Trial: Each repetition of the experiment. Tossing a coin once = one trial. Rolling a die 50 times = 50 trials.
- Outcome: A possible result of a single trial. For a coin: {Head, Tail}. For a die: {1, 2, 3, 4, 5, 6}.
- Sample Space (S): The set of all possible outcomes of an experiment. For a coin, S = {H, T}.
- Event: A subset of the sample space. "Getting an even number on a die" = {2, 4, 6}.
- Favourable Outcome: An outcome that satisfies the condition of the event.
Fundamental properties of probability:
- For any event E: 0 ≤ P(E) ≤ 1.
- P(E) = 0 means the event is impossible (it never occurred in the experiment).
- P(E) = 1 means the event is certain (it occurred in every trial).
- P(E) + P(not E) = 1 — the event and its complement always sum to 1.
- The sum of probabilities of all possible outcomes = 1.
- Probability can be expressed as a fraction (3/10), a decimal (0.3), or a percentage (30%).
Probability - Experimental Approach Formula
Key Formulas:
1. Experimental Probability:
P(E) = Number of favourable trials / Total number of trials
2. Complementary Event:
P(not E) = 1 − P(E)
3. Sum of all probabilities:
- P(E₁) + P(E₂) + ... + P(Eₙ) = 1
- where E₁, E₂, ..., Eₙ are all possible outcomes.
4. Range of probability:
- 0 ≤ P(E) ≤ 1
5. Expressing probability:
- As a fraction (3/10), decimal (0.3), or percentage (30%)
Derivation and Proof
How Experimental Probability Works:
Step 1: Define the experiment
- Choose a random experiment (e.g., tossing a coin, rolling a die, drawing a card).
Step 2: Conduct the experiment
- Perform the experiment a fixed number of times (the trials).
- The more trials, the more reliable the result.
Step 3: Record the outcomes
- Note down the result of each trial.
- Count the number of times each outcome occurs.
Step 4: Calculate the probability
- Count the number of favourable outcomes (the event you are interested in).
- Divide by the total number of trials.
- P(E) = favourable / total
Step 5: Interpret the result
- The result is an approximation that improves as the number of trials increases.
- With very many trials, experimental probability approaches the theoretical probability.
Example:
- A coin is tossed 100 times. Heads appears 47 times.
- P(Heads) = 47/100 = 0.47
- P(Tails) = 1 − 0.47 = 0.53
Types and Properties
Types of probability experiments in Class 9:
1. Coin Tossing
- Outcomes: Heads (H) or Tails (T)
- P(H) = Number of heads / Total tosses
2. Rolling a Die
- Outcomes: 1, 2, 3, 4, 5, 6
- P(any number) = Number of times it appeared / Total rolls
3. Drawing from a Bag
- Balls or objects of different colours are drawn randomly.
- After each draw, record the colour, replace the ball, and draw again.
4. Survey-based Probability
- Data collected from surveys: blood groups, ages, preferences, etc.
- P(event) = Number of people with that attribute / Total people surveyed
5. Sports and Games
- Win/loss records of teams or players.
- P(win) = Number of wins / Total matches
6. Quality Control
- Defective vs non-defective items in manufacturing.
- P(defective) = Number of defective items / Total items checked
Solved Examples
Example 1: Example 1: Coin tossing experiment
Problem: A coin is tossed 200 times. Heads appears 112 times. Find P(Heads) and P(Tails).
Solution:
Given:
- Total trials = 200, Heads = 112
Calculating:
- P(Heads) = 112/200 = 0.56
- P(Tails) = 1 − 0.56 = 0.44
Verification: 0.56 + 0.44 = 1 ✓
Answer: P(Heads) = 0.56; P(Tails) = 0.44.
Example 2: Example 2: Die rolling experiment
Problem: A die is rolled 60 times with results: 1 appears 8 times, 2 appears 12 times, 3 appears 10 times, 4 appears 9 times, 5 appears 11 times, 6 appears 10 times. Find the probability of getting: (a) a 3, (b) an even number.
Solution:
Given:
- Total rolls = 60
(a) P(3):
- P(3) = 10/60 = 1/6
(b) P(even number):
- Even numbers: 2 (12 times) + 4 (9 times) + 6 (10 times) = 31
- P(even) = 31/60 ≈ 0.517
Answer: P(3) = 1/6; P(even number) ≈ 0.517.
Example 3: Example 3: Blood group survey
Problem: In a survey of 1000 people, the blood groups were: O — 400, A — 230, B — 270, AB — 100. Find the probability that a randomly selected person has blood group B.
Solution:
Given:
- Total = 1000, Blood group B = 270
Calculating:
- P(B) = 270/1000 = 0.27
Answer: P(blood group B) = 0.27 or 27%.
Example 4: Example 4: Quality control
Problem: Out of 500 bulbs tested, 15 were found defective. Find the probability that a bulb picked at random is (a) defective, (b) not defective.
Solution:
Given:
- Total = 500, Defective = 15
(a) P(defective):
- P(defective) = 15/500 = 0.03
(b) P(not defective):
- P(not defective) = 1 − 0.03 = 0.97
Answer: P(defective) = 0.03; P(not defective) = 0.97.
Example 5: Example 5: Cricket match record
Problem: A cricket team has played 40 matches. They won 24, lost 12, and drew 4. Find the probability of (a) winning, (b) losing, (c) drawing.
Solution:
Given:
- Total = 40, Won = 24, Lost = 12, Drawn = 4
Calculating:
- P(win) = 24/40 = 3/5 = 0.6
- P(loss) = 12/40 = 3/10 = 0.3
- P(draw) = 4/40 = 1/10 = 0.1
Verification: 0.6 + 0.3 + 0.1 = 1 ✓
Answer: P(win) = 0.6; P(loss) = 0.3; P(draw) = 0.1.
Example 6: Example 6: Drawing balls from a bag
Problem: A bag contains red, blue, and green balls. In 80 draws (with replacement), red was drawn 28 times, blue 32 times, and green 20 times. Find the probability of drawing each colour.
Solution:
Given:
- Total = 80
Calculating:
- P(Red) = 28/80 = 7/20 = 0.35
- P(Blue) = 32/80 = 2/5 = 0.4
- P(Green) = 20/80 = 1/4 = 0.25
Verification: 0.35 + 0.4 + 0.25 = 1 ✓
Answer: P(Red) = 0.35; P(Blue) = 0.4; P(Green) = 0.25.
Example 7: Example 7: Weather prediction
Problem: Over the past 365 days, it rained on 73 days. Find the probability that it rains on a given day, and the probability that it does NOT rain.
Solution:
Given:
- Total days = 365, Rainy days = 73
Calculating:
- P(rain) = 73/365 = 1/5 = 0.2
- P(no rain) = 1 − 0.2 = 0.8
Answer: P(rain) = 0.2; P(no rain) = 0.8.
Example 8: Example 8: Student survey
Problem: In a class of 50 students, 15 like cricket, 20 like football, 10 like badminton, and 5 like tennis. A student is picked at random. Find the probability that the student likes football.
Solution:
Given:
- Total = 50, Football = 20
Calculating:
- P(football) = 20/50 = 2/5 = 0.4
Answer: P(student likes football) = 0.4 or 40%.
Example 9: Example 9: Two-coin toss
Problem: Two coins are tossed simultaneously 150 times. The results are: 2 Heads — 38 times, 1 Head — 76 times, 0 Heads — 36 times. Find the probability of getting at least one head.
Solution:
Given:
- Total = 150
- At least 1 head = 2H (38) + 1H (76) = 114
Calculating:
- P(at least 1 head) = 114/150 = 19/25 = 0.76
Answer: P(at least one head) = 0.76.
Example 10: Example 10: Verifying complementary events
Problem: In a bag, 5 red and 3 blue balls are there. In 400 draws (with replacement), a red ball was drawn 245 times. Find P(red) and verify P(red) + P(blue) = 1.
Solution:
Given:
- Total = 400, Red = 245
Calculating:
- P(Red) = 245/400 = 0.6125
- P(Blue) = 1 − 0.6125 = 0.3875
- P(Red) + P(Blue) = 0.6125 + 0.3875 = 1 ✓
Answer: P(Red) = 0.6125; P(Blue) = 0.3875. Sum = 1. Verified.
Real-World Applications
Applications of Experimental Probability:
- Weather Forecasting: When a weather report says "70% chance of rain tomorrow", this is based on historical data. Meteorologists analysed past days with similar atmospheric conditions and found that it rained on 70% of such days. This is experimental probability applied to weather prediction. Decades of temperature, humidity, and pressure data are used to generate these forecasts.
- Insurance and Risk Assessment: Insurance companies collect massive amounts of data on accidents, health events, and natural disasters. By calculating the experimental probability of claims from historical records, they set premium rates that balance risk and profitability. Higher probability of claims (for example, for older drivers or flood-prone areas) leads to higher premiums. The entire insurance industry is built on probability calculations.
- Medical Research and Drug Testing: When a new drug or vaccine is tested, researchers conduct clinical trials with thousands of patients. If 9,200 out of 10,000 vaccinated individuals did not get the disease while 500 out of 10,000 unvaccinated individuals got it, the experimental probability helps measure effectiveness. Regulatory agencies like the FDA approve drugs based on these probability calculations.
- Quality Control in Manufacturing: Factories test random samples from each production batch. If 12 out of 2,000 tested bulbs are defective, P(defective) = 12/2000 = 0.006 or 0.6%. The batch is accepted if this is below the Acceptable Quality Limit (AQL). This prevents defective products from reaching consumers and helps maintain brand reputation.
- Sports Analytics and Strategy: Batting averages in cricket (total runs divided by total innings), free-throw percentages in basketball, and goal-scoring rates in football are all experimental probabilities. Coaches and team analysts use these statistics for player selection, match strategy, and performance evaluation. Modern sports teams employ entire data analytics departments.
- Genetics and Heredity: Gregor Mendel’s experiments with pea plants in the 1860s used experimental probability to discover the laws of inheritance. By recording the frequency of traits (tall vs short, yellow vs green) across thousands of plants, he determined the probability ratios that govern heredity. Modern genetics continues to use probability extensively.
- Market Research and Business: Companies conduct surveys and focus groups to estimate consumer preferences. If 680 out of 1,000 surveyed customers preferred Product A, P(preference) = 0.68. This experimental probability guides decisions about product development, marketing budget allocation, and inventory management.
- Traffic Engineering and Road Safety: Traffic planners collect data on accident rates, traffic volume, and signal timings at intersections. The probability of accidents at particular locations determines where safety improvements (speed bumps, traffic signals, guardrails) are needed. Experimental probability from years of data saves lives.
Key Points to Remember
- Experimental probability = (Number of times event occurred) / (Total number of trials). This is also called empirical probability or relative frequency.
- The value of probability always lies between 0 and 1 (inclusive). P(E) = 0 means the event is impossible; P(E) = 1 means it is certain.
- P(E) + P(not E) = 1 — an event and its complementary event always have probabilities that sum to 1. This is the most useful shortcut in probability calculations.
- The sum of probabilities of all possible outcomes of an experiment = 1. For a die: P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1.
- Experimental probability may differ from theoretical probability, but they converge as the number of trials increases. This is the Law of Large Numbers.
- An experiment (or random experiment) is any process whose outcome is uncertain. A trial is one instance of the experiment.
- The sample space is the set of all possible outcomes. For a coin: S = {H, T}. For a die: S = {1, 2, 3, 4, 5, 6}.
- An event is a subset of the sample space. An elementary event has exactly one outcome; a compound event has multiple outcomes.
- Experimental probability can be expressed as a fraction, decimal, or percentage. For example, P = 3/10 = 0.3 = 30%.
- Different repetitions of the same experiment may give different experimental probabilities. This variability decreases as the number of trials increases.
- Probability was formally studied starting in the 17th century by Pascal and Fermat, who analysed games of chance.
- Class 9 focuses on experimental probability from observed data. Theoretical probability (using equally likely outcomes) is studied in Class 10.
Practice Problems
- A coin is tossed 500 times. Tails appears 237 times. Find P(Heads) and P(Tails).
- A die is thrown 120 times. The number 5 appeared 22 times. Find the probability of getting a 5.
- In 200 births at a hospital, 108 were boys and 92 were girls. Find P(girl).
- A factory produced 2000 items. 50 were defective. Find the probability of picking a non-defective item.
- Three coins are tossed 300 times. All three heads appeared 40 times. Find P(all three heads).
- A bag contains red and black balls. In 150 draws (with replacement), red appeared 90 times. Find P(red) and P(black).
- Out of 1000 families surveyed, 460 had exactly 2 children. Find the probability that a randomly selected family has exactly 2 children.
- In 250 cricket matches, a team won the toss 130 times. Find the probability of winning the toss.
Frequently Asked Questions
Q1. What is experimental probability?
Experimental probability is the ratio of the number of times an event occurs to the total number of trials conducted. It is based on actual observations from experiments.
Q2. How is experimental probability different from theoretical probability?
Experimental probability is based on actual trial results. Theoretical probability is calculated using logic and equally likely outcomes (e.g., P(Head) = 1/2 for a fair coin). As trials increase, experimental probability approaches theoretical probability.
Q3. Can probability be negative?
No. Probability always lies between 0 and 1 (inclusive). P(E) = 0 means impossible, P(E) = 1 means certain.
Q4. What is a complementary event?
The complement of event E (written as not E or E') consists of all outcomes that are NOT in E. P(not E) = 1 − P(E).
Q5. Why does experimental probability change each time?
Because the results of random experiments vary each time. Different sets of trials may give different outcomes. With more trials, the results stabilise and converge towards the theoretical probability.
Q6. Is experimental probability always a fraction?
It can be expressed as a fraction, decimal, or percentage. For example, 3/10 = 0.3 = 30%. All forms are correct.
Q7. What is the formula for experimental probability?
P(E) = Number of trials in which E occurs / Total number of trials.
Q8. Is probability covered in CBSE Class 9?
Yes. The experimental (empirical) approach to probability is part of CBSE Class 9 Mathematics, Chapter: Probability.
Q9. What is a trial in probability?
A trial is a single performance of an experiment. For example, one toss of a coin is one trial, one roll of a die is one trial. Multiple trials of the same experiment form the basis for calculating experimental probability.
Q10. Can the sum of probabilities of all outcomes exceed 1?
No. The sum of probabilities of all possible outcomes of an experiment is always exactly 1. This is because the outcomes together cover the entire sample space.
