Probability of an Event
Probability measures the likelihood of an event occurring. In Class 9, probability is introduced through the experimental (or empirical) approach, where the probability is estimated based on the results of actual experiments or observations.
The probability of an event is a number between 0 and 1. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain.
This topic builds on the concepts of experiments, outcomes, and events to establish the formula for experimental probability.
What is Probability of an Event?
Definition: The probability of an event E is the ratio of the number of trials in which the event occurred to the total number of trials performed.
P(E) = Number of trials in which E occurred / Total number of trials
Key Terms:
- Experiment: An action or process that leads to well-defined outcomes.
- Outcome: A possible result of an experiment.
- Event: A collection of one or more outcomes.
- Trial: Each repetition of an experiment.
Range of Probability:
- 0 ≤ P(E) ≤ 1 for any event E.
- P(E) = 0 ⇒ impossible event
- P(E) = 1 ⇒ certain event
Probability of an Event Formula
Key Formulas:
1. Experimental Probability:
P(E) = Number of favourable trials / Total number of trials
2. Complementary Events:
P(E) + P(not E) = 1
3. Probability bounds:
- 0 ≤ P(E) ≤ 1
- Sum of probabilities of all outcomes = 1
Derivation and Proof
Understanding Experimental Probability:
Step 1: Perform the experiment a large number of times (n trials).
Step 2: Count the number of times the event E occurs (favourable outcomes = m).
Step 3: Calculate P(E) = m/n.
Example Process:
- Toss a coin 100 times.
- Suppose heads appears 48 times.
- P(heads) = 48/100 = 0.48
- P(tails) = 52/100 = 0.52
- P(heads) + P(tails) = 0.48 + 0.52 = 1 ✓
Important:
- As the number of trials increases, experimental probability approaches the theoretical probability.
- This is known as the Law of Large Numbers.
Types and Properties
Types of Events:
1. Certain Event
- An event that always occurs. P(E) = 1.
- Example: Getting a number less than 7 when rolling a standard die.
- An event that never occurs. P(E) = 0.
- Example: Getting the number 8 when rolling a standard die.
3. Equally Likely Events
- Events that have the same probability of occurring.
- Example: Getting heads or tails in a fair coin toss.
- Two events E and E′ such that P(E) + P(E′) = 1.
Solved Examples
Example 1: Example 1: Coin toss experiment
Problem: A coin is tossed 200 times. Heads appeared 112 times. Find P(heads) and P(tails).
Solution:
- P(heads) = 112/200 = 0.56
- P(tails) = 88/200 = 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(8), 2(12), 3(10), 4(9), 5(11), 6(10). Find P(2) and P(even number).
Solution:
- P(2) = 12/60 = 1/5 = 0.2
- Even: 2, 4, 6. Frequency = 12 + 9 + 10 = 31.
- P(even) = 31/60 ≈ 0.517
Answer: P(2) = 0.2; P(even) ≈ 0.517.
Example 3: Example 3: Survey data
Problem: In a survey of 500 families, 300 have 2 children. Find P(family has 2 children).
Solution:
- P(2 children) = 300/500 = 3/5 = 0.6
Answer: P(2 children) = 0.6.
Example 4: Example 4: Defective items
Problem: In a batch of 400 bulbs, 16 are defective. Find P(defective) and P(not defective).
Solution:
- P(defective) = 16/400 = 0.04
- P(not defective) = 1 − 0.04 = 0.96
Answer: P(defective) = 0.04; P(not defective) = 0.96.
Example 5: Example 5: Cricket batting data
Problem: A cricketer scored: 45, 0, 78, 32, 120, 0, 55, 18, 0, 62 in 10 innings. Find P(score 0) and P(more than 50).
Solution:
- P(0) = 3/10 = 0.3
- Innings > 50: 78, 120, 55, 62 = 4. P(>50) = 4/10 = 0.4
Answer: P(0) = 0.3; P(>50) = 0.4.
Example 6: Example 6: Weather data
Problem: It rained on 45 out of 365 days. Find P(rain) and P(no rain).
Solution:
- P(rain) = 45/365 = 9/73 ≈ 0.123
- P(no rain) = 1 − 9/73 ≈ 0.877
Answer: P(rain) ≈ 0.123; P(no rain) ≈ 0.877.
Example 7: Example 7: Impossible and certain events
Problem: A die is rolled. Find P(number less than 7) and P(getting 8).
Solution:
- P(less than 7) = 6/6 = 1 (certain event)
- P(8) = 0/6 = 0 (impossible event)
Answer: P(less than 7) = 1; P(8) = 0.
Example 8: Example 8: Blood group data
Problem: 250 donors: A(80), B(65), AB(30), O(75). Find P(B), P(AB), P(not O).
Solution:
- P(B) = 65/250 = 0.26
- P(AB) = 30/250 = 0.12
- P(not O) = 1 − 75/250 = 0.7
Answer: P(B) = 0.26; P(AB) = 0.12; P(not O) = 0.7.
Real-World Applications
Applications:
- Weather Forecasting: Probability of rain from historical data.
- Quality Control: Estimating defective item rates in production.
- Insurance: Premiums based on probability of claims.
- Sports: Win probabilities from past performance.
- Medicine: Probability of disease given symptoms aids diagnosis.
Key Points to Remember
- Experimental probability is based on actual experiments.
- P(E) = favourable trials / total trials.
- Probability ranges from 0 to 1.
- P(E) = 0 means impossible; P(E) = 1 means certain.
- P(E) + P(not E) = 1.
- Sum of probabilities of all outcomes = 1.
- More trials → closer to theoretical probability (Law of Large Numbers).
- Experimental probability can differ between experimenters.
- This is covered in NCERT Class 9, Chapter 15 (Probability).
Practice Problems
- A coin is tossed 500 times. Tails appears 265 times. Find P(heads).
- In a factory, 12 out of 600 items are defective. Find P(defective).
- A die is thrown 120 times. The number 5 appeared 22 times. Find P(5).
- In a class of 40, 28 like cricket and 12 like football. Find P(likes cricket).
- In 80 draws, a red ball was drawn 24 times. Find P(red ball).
- P(event) = 0.35. Find P(event does not occur).
Frequently Asked Questions
Q1. What is the probability of an event?
The probability of an event is a number between 0 and 1 that measures how likely the event is to occur. It is calculated as favourable outcomes divided by total outcomes.
Q2. What is the difference between experimental and theoretical probability?
Experimental probability is based on actual results. Theoretical probability is calculated mathematically. As trials increase, experimental probability approaches theoretical probability.
Q3. What is a certain event?
A certain event always occurs. Its probability is 1. Example: rolling a number less than 7 on a standard die.
Q4. What is an impossible event?
An impossible event never occurs. Its probability is 0. Example: rolling a 7 on a standard die.
Q5. Can probability be negative or greater than 1?
No. Probability is always between 0 and 1, inclusive.
Q6. What are complementary events?
Two events E and not-E are complementary if P(E) + P(not E) = 1. One occurs exactly when the other does not.
Q7. Why do different people get different experimental probabilities?
Because experimental probability depends on the specific trials performed. With more trials, results converge to the theoretical value.
Q8. Is this in the CBSE Class 9 syllabus?
Yes. Probability is covered in Chapter 15 of NCERT Class 9 Mathematics, focusing on the experimental approach.
Related Topics
- Probability - Experimental Approach
- Complementary Events in Probability
- Theoretical Probability
- Experimental Probability
- Probability - Theoretical Approach
- Probability of Simple Events
- Probability of Compound Events
- Impossible and Sure Events
- Probability with Playing Cards
- Probability with Dice
- Probability with Coins
