Class 9 Mathematics Notes Chapter 16 (Chapter 16) – Examplar Problem (Englisha) Book

Alright class, let's focus on Chapter 16, which deals with Probability. This is a crucial topic, not just for your Class 9 understanding but also forms a base for many competitive government exams. We'll be looking at it from the perspective of the NCERT Exemplar, which helps build a stronger conceptual foundation.

Chapter 16: Probability (Based on NCERT Class 9 Exemplar)

1. Understanding Probability: The Empirical Approach

• In Class 9, we primarily deal with Empirical Probability (also called Experimental Probability).
• This is based on the results of actual experiments or observations.
• We perform an activity (like tossing a coin or rolling a die) multiple times and observe the outcomes.
• Definition: The empirical probability P(E) of an event E happening is calculated as:
  P(E) = (Number of trials in which the event E happened) / (Total number of trials)

2. Key Terminology

• Trial: A single performance of an experiment (e.g., one toss of a coin, one roll of a die).
• Experiment: An action or process that results in well-defined outcomes (e.g., tossing a coin 100 times, rolling a die 50 times).
• Outcome: A possible result of a trial (e.g., getting 'Heads' on a coin toss, rolling a '4' on a die).
• Event: A specific outcome or a collection of outcomes from an experiment that we are interested in (e.g., the event of getting an even number when rolling a die, the event of getting at least one head when tossing two coins).

3. Calculating Empirical Probability: Examples

• Coin Toss: If a coin is tossed 500 times and 'Heads' appears 280 times, the empirical probability of getting a Head is:
  P(Head) = 280 / 500 = 28 / 50 = 14 / 25 = 0.56
  The empirical probability of getting a Tail is:
  P(Tail) = (Total Tosses - Head Count) / Total Tosses = (500 - 280) / 500 = 220 / 500 = 22 / 50 = 11 / 25 = 0.44
• Dice Roll: A die is rolled 200 times. The frequency of outcomes (1, 2, 3, 4, 5, 6) is recorded. If the number '5' appeared 35 times, the empirical probability of getting a '5' is:
  P(Getting a 5) = 35 / 200 = 7 / 40 = 0.175
• Surveys/Data: If a survey of 150 families shows that 60 families have 2 girls, the probability of randomly selecting a family with 2 girls is:
  P(Family with 2 girls) = 60 / 150 = 6 / 15 = 2 / 5 = 0.4

4. Important Properties of Probability

• Range: The probability of any event E always lies between 0 and 1, inclusive.
  0 ≤ P(E) ≤ 1
  • Probability cannot be negative.
  • Probability cannot be greater than 1.
• Impossible Event: An event that cannot happen under any circumstances has a probability of 0. (e.g., getting a '7' when rolling a standard six-sided die).
• Sure Event (or Certain Event): An event that is guaranteed to happen has a probability of 1. (e.g., getting a number less than 7 when rolling a standard six-sided die).
• Sum of Probabilities: In an experiment, the sum of the probabilities of all possible distinct outcomes is always 1.
  For the coin toss example above: P(Head) + P(Tail) = 0.56 + 0.44 = 1.00

5. Focus Areas for Exams (Based on Exemplar)

• Interpreting Data: Many questions involve reading data from tables (like frequency distributions) or graphs and calculating probabilities based on that data. Pay close attention to the 'Total Number of Trials'.
• Combined Events: Calculating the probability of events like 'getting an even number', 'getting a prime number', 'getting a number greater than 4' from dice roll data. You need to sum up the frequencies of the relevant outcomes first.
• Real-world Scenarios: Problems related to weather forecasts, manufacturing defects, survey results, etc., where probability is calculated based on past observations.

Key Takeaway: Empirical probability is about observed frequencies in an experiment. The more trials you conduct, the closer the empirical probability might get to the theoretical probability (which you'll study in more detail later). For Class 9 and related exams, master the formula and its application to data sets.

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on Class 9 Probability concepts, suitable for practice:

1. A coin is tossed 1000 times with the following frequencies: Head: 455, Tail: 545. The empirical probability of getting a Head is:
   a) 0.45
   b) 0.545
   c) 0.455
   d) 1

2. A die is thrown 300 times and the outcomes are noted as follows:
   Outcome: | 1 | 2 | 3 | 4 | 5 | 6 |
   Frequency:| 60| 50| 45| 40| 55| 50|
   The probability of getting an even number is:
   a) 140/300
   b) 150/300
   c) 160/300
   d) 145/300

3. Which of the following cannot be the empirical probability of an event?
   a) 2/3
   b) -0.5
   c) 15%
   d) 0.7

4. In a survey of 200 students, 80 were found to like coffee. If a student is chosen at random, what is the probability that they do not like coffee?
   a) 80/200
   b) 100/200
   c) 120/200
   d) 1

5. The probability of an event that is certain to happen is:
   a) 0
   b) 1/2
   c) 1
   d) Greater than 1

6. A bag contains 5 red balls and 3 black balls. A ball is drawn at random. What is the probability of drawing a black ball based on this composition (assuming this reflects experimental probability)?
   a) 5/8
   b) 3/8
   c) 3/5
   d) 1

7. In an experiment, the sum of the probabilities of all possible distinct outcomes is always:
   a) 0
   b) 1
   c) Less than 1
   d) More than 1

8. A weather station's data shows that out of the past 250 consecutive days, its forecast was correct 175 times. The probability that on a given day the forecast was not correct is:
   a) 175/250
   b) 1
   c) 0
   d) 75/250

9. If P(E) = 0.38, what is the probability of 'not E'?
   a) 0.38
   b) 0.62
   c) 0.72
   d) 1

10. A spinner has 4 equal sectors coloured yellow, blue, green, and red. After spinning it 100 times, the frequencies are: Yellow: 20, Blue: 30, Green: 25, Red: 25. What is the empirical probability of not landing on Blue?
    a) 30/100
    b) 75/100
    c) 70/100
    d) 25/100

Answers to MCQs:

1. c) 0.455 (Calculation: 455 / 1000)
2. b) 150/300 (Even numbers are 2, 4, 6. Frequencies: 50 + 40 + 50 = 140. Wait, check calculation: 50+40+50 = 140. Option b is 150/300. Let me re-read the frequencies. 60, 50, 45, 40, 55, 50. Even frequencies: 50 (for 2) + 40 (for 4) + 50 (for 6) = 140. The probability is 140/300. Option (a) is 140/300. Let me correct the answer key. Correction: Answer should be (a). Rechecking options: (a) 140/300, (b) 150/300, (c) 160/300, (d) 145/300. Yes, the sum is 140. So the probability is 140/300. The answer is (a). Self-correction: Updating the answer key.)
   Corrected Answer: a) 140/300
3. b) -0.5 (Probability cannot be negative)
4. c) 120/200 (Students who do not like coffee = 200 - 80 = 120. Probability = 120/200)
5. c) 1 (Definition of a sure event)
6. b) 3/8 (Total balls = 5 + 3 = 8. Black balls = 3. Probability = 3/8)
7. b) 1 (Fundamental property of probability)
8. d) 75/250 (Number of incorrect forecasts = 250 - 175 = 75. Probability = 75/250)
9. b) 0.62 (P(not E) = 1 - P(E) = 1 - 0.38 = 0.62)
10. c) 70/100 (Total trials = 100. Frequency of Blue = 30. Frequency of not Blue = 100 - 30 = 70. Probability = 70/100)

Study these notes carefully and practice similar problems from your Exemplar book. Let me know if any concept is unclear!