Class 9 Mathematics Notes Chapter 15 (Chapter 15) – Examplar Problem (Englisha) Book

Alright class, let's get started with Chapter 15, Probability, from your NCERT Exemplar. This is a crucial chapter, not just for your Class 9 exams, but also because probability concepts frequently appear in various government exams. We'll focus on the core ideas and how to apply them, especially the experimental or empirical approach relevant to your syllabus.

Chapter 15: Probability - Detailed Notes for Competitive Exams

1. Introduction & Basic Terminology

• Experiment: An action or process that results in one or several well-defined outcomes.
  • Examples: Tossing a coin, rolling a die, drawing a card from a deck, observing the number of defective items produced in a factory lot.
• Trial: A single performance of an experiment.
  • Example: Tossing a coin once is a trial. Rolling a die once is a trial.
• Outcome: A possible result of a trial/experiment.
  • Examples: Getting 'Heads' when tossing a coin; getting a '4' when rolling a die.
• Event (E): A collection of one or more outcomes of an experiment. We usually denote events with capital letters like E, A, B, etc.
  • Examples: Getting an even number (outcomes: 2, 4, 6) when rolling a die; getting at least one head when tossing two coins (outcomes: HT, TH, HH).
• Empirical Probability (or Experimental Probability): This is the type of probability focused on in Class 9. It is based on the results of an actual experiment conducted repeatedly. It's calculated based on observations.

2. Calculating Empirical Probability

The empirical probability of an event E happening, denoted by P(E), is calculated as:

P(E) = (Number of trials in which the event E happened) / (Total number of trials)

• Key Idea: This probability is based on past data or the results of performing an experiment many times. As the number of trials increases, the empirical probability tends to get closer to the theoretical probability (which you'll study more in Class 10).
• Example 1 (Coin Toss): A coin is tossed 500 times. Heads appear 280 times, and tails appear 220 times.
  • Total number of trials = 500
  • Let H be the event of getting a Head. Number of trials where H happened = 280.
    P(H) = 280 / 500 = 28 / 50 = 14 / 25 = 0.56
  • Let T be the event of getting a Tail. Number of trials where T happened = 220.
    P(T) = 220 / 500 = 22 / 50 = 11 / 25 = 0.44
• Example 2 (Dice Roll): A die is thrown 100 times. The frequency of outcomes 1, 2, 3, 4, 5, 6 are recorded as 15, 20, 18, 12, 25, 10 respectively.
  • Total number of trials = 100
  • Let E be the event of getting an odd number (outcomes 1, 3, 5).
    Number of trials where E happened = Frequency of 1 + Frequency of 3 + Frequency of 5 = 15 + 18 + 25 = 58
    P(E) = 58 / 100 = 0.58
  • Let F be the event of getting a number greater than 4 (outcomes 5, 6).
    Number of trials where F happened = Frequency of 5 + Frequency of 6 = 25 + 10 = 35
    P(F) = 35 / 100 = 0.35

3. Properties of Probability

These properties hold true for any type of probability, including empirical probability based on sufficient trials.

• Range of Probability: 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. The probability of an impossible event is 0.
  • Example: Getting a '7' when rolling a standard six-sided die. P(getting a 7) = 0.
• Sure Event (or Certain Event): An event that is certain to happen. The probability of a sure event is 1.
  • Example: Getting a number less than 7 when rolling a standard six-sided die. P(number < 7) = 1.
• Sum of Probabilities: For an experiment, the sum of the probabilities of all possible elementary events (events with only one outcome) is always 1.
  • Example (Coin Toss from above): P(H) + P(T) = 0.56 + 0.44 = 1.00
  • Example (Dice Roll from above): P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = (15/100) + (20/100) + (18/100) + (12/100) + (25/100) + (10/100) = (15+20+18+12+25+10)/100 = 100/100 = 1.

4. Applications in Government Exams

Questions in competitive exams often involve interpreting data presented in tables or paragraphs and then calculating empirical probability.

• Surveys: Data from surveys (e.g., number of families with certain numbers of vehicles, opinions on a policy) can be used to ask probability questions.
  • Example: A survey of 200 families found 50 families have no vehicles, 90 have 1 vehicle, 40 have 2 vehicles, and 20 have more than 2 vehicles. If a family is chosen at random, what is the probability that they have exactly 1 vehicle?
    • Total families (trials) = 200
    • Families with 1 vehicle (favourable event) = 90
    • P(exactly 1 vehicle) = 90 / 200 = 9 / 20 = 0.45
• Quality Control: Data on defective/non-defective items produced.
  • Example: In a batch of 1000 bulbs, 45 were found defective. What is the probability that a bulb selected at random is non-defective?
    • Total bulbs (trials) = 1000
    • Defective bulbs = 45
    • Non-defective bulbs = 1000 - 45 = 955
    • P(non-defective) = 955 / 1000 = 0.955
• Frequency Distribution Tables: Data is often given in frequency tables (like the dice example above). You need to read the table correctly to find the number of favourable outcomes and the total number of trials.

5. Important Points to Remember

• Always identify the total number of trials first. This is your denominator.
• Clearly identify the event for which you need to calculate the probability.
• Count the number of trials where the specific event occurred. This is your numerator.
• Ensure your final probability is between 0 and 1.
• Answers can be expressed as fractions (in simplest form), decimals, or sometimes percentages (though fractions and decimals are more common in probability calculations).

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on Class 9 Probability concepts, keeping competitive exams in mind:

1. A coin is tossed 1000 times with the following frequencies: Head: 455, Tail: 545. The empirical probability of getting a Head is:
   (a) 0.545
   (b) 0.455
   (c) 4.55
   (d) 5.45

2. In a cricket match, a batswoman hits a boundary 6 times out of 30 balls she plays. Find the probability that she did not hit a boundary.
   (a) 0.2
   (b) 0.8
   (c) 0.6
   (d) 0.4

3. A die is thrown 200 times. The frequency of the outcome '3' is 40. The probability of getting the outcome '3' is:
   (a) 1/5
   (b) 1/4
   (c) 3/10
   (d) 4/5

4. Which of the following cannot be the empirical probability of an event?
   (a) 3/4
   (b) 0.01
   (c) 1.001
   (d) 5/6

5. In a survey of 50 students, 15 like coffee and 35 like tea. A student is selected at random. What is the probability that the selected student likes coffee?
   (a) 3/10
   (b) 7/10
   (c) 3/7
   (d) 7/3

6. A bag contains 5 red balls and 3 blue balls. An experiment consists of picking a ball, noting its colour, and replacing it. This is done 80 times. If red balls were picked 45 times, what is the empirical probability of picking a blue ball?
   (a) 45/80
   (b) 35/80
   (c) 3/8
   (d) 5/8

7. The record of a weather station shows that out of the past 250 consecutive days, its weather forecasts were correct 175 times. What is the probability that on a given day it was not correct?
   (a) 175/250
   (b) 7/10
   (c) 3/10
   (d) 1/4

8. An organisation selected 2400 families at random and surveyed them to determine a relationship between income level and the number of vehicles in a family. The data is tabulated below:

   Monthly Income (in Rs)	Vehicles per family (0)	Vehicles per family (1)	Vehicles per family (2)	Vehicles per family (Above 2)
   Less than 7000	10	160	25	0
   7000-10000	0	305	27	2
   10000-13000	1	535	29	1
   13000-16000	2	469	59	25
   16000 or more	1	579	82	88

   What is the probability that a randomly chosen family earns Rs 10000-13000 per month and owns exactly 2 vehicles?
   (a) 29/566
   (b) 1/80
   (c) 29/2400
   (d) 535/2400

9. Using the table from Q8, what is the probability that a randomly chosen family owns not more than 1 vehicle?
   (a) (10+0+1+2+1) / 2400
   (b) (160+305+535+469+579) / 2400
   (c) (14 + 2048) / 2400
   (d) 2062 / 2400

10. If P(E) = 0.37, what is the probability of 'not E'?
    (a) 0.37
    (b) 0.63
    (c) 0
    (d) 1

Answers to MCQs:

1. (b) 0.455 [Calculation: 455 / 1000]
2. (b) 0.8 [Calculation: Boundary hit 6 times, Not hit = 30 - 6 = 24 times. P(Not hit) = 24 / 30 = 4/5 = 0.8]
3. (a) 1/5 [Calculation: 40 / 200 = 1/5]
4. (c) 1.001 [Reason: Probability cannot be greater than 1]
5. (a) 3/10 [Calculation: 15 / 50 = 3/10]
6. (b) 35/80 [Calculation: Total trials = 80. Red picked = 45. Blue picked = 80 - 45 = 35. P(Blue) = 35/80 = 7/16. The initial composition (5 red, 3 blue) is extra info for empirical probability calculation here.]
7. (c) 3/10 [Calculation: Total days = 250. Correct forecasts = 175. Incorrect forecasts = 250 - 175 = 75. P(Not correct) = 75 / 250 = 3/10]
8. (c) 29/2400 [Calculation: Favourable outcomes (Family earns 10000-13000 AND owns 2 vehicles) = 29. Total families = 2400. P = 29/2400]
9. (d) 2062 / 2400 [Calculation: 'Not more than 1' means 0 vehicles OR 1 vehicle. Total families with 0 vehicles = 10+0+1+2+1 = 14. Total families with 1 vehicle = 160+305+535+469+579 = 2048. Total favourable = 14 + 2048 = 2062. P = 2062 / 2400]
10. (b) 0.63 [Calculation: P(not E) = 1 - P(E) = 1 - 0.37 = 0.63]

Make sure you understand the reasoning behind each answer. Practice more problems from the Exemplar book and other sources focusing on interpreting data and applying the basic probability formula. Good luck!