Probability Chapter Notes | Mathematics Class 8 ICSE PDF Download

Introduction

Probability is a way to measure how likely something is to happen. We often use words like "chance," "possible," or "likely" in everyday life without fully understanding their meaning. In mathematics, probability helps us understand the uncertainty of events by looking at what has happened in experiments. This chapter introduces the concept of probability, focusing on random experiments and their outcomes, and explains how to calculate the chance of specific events occurring.

What is Probability?

• Probability is about understanding the chance of an event happening.
• Words like possible, probable, chance, and likely show uncertainty in events.
• Probability is based on real outcomes of experiments.
• An experiment is an action with clear outcomes.
• A random experiment has multiple possible outcomes, and we cannot predict the exact outcome in advance.
• In this chapter, "experiment" refers to a random experiment.
• Examples of random experiments include:
  • Tossing a coin (outcomes: head or tail).
  • Throwing a die (outcomes: 1, 2, 3, 4, 5, 6).
  • Drawing a card from a deck of 52 cards.
• In a random experiment, all possible outcomes are known, but the specific outcome is unpredictable.
• Example: Throwing a die is a random experiment because we know the outcomes are 1, 2, 3, 4, 5, or 6, but we cannot say which number will appear.
• Example 1: Write all possible outcomes when:
• (i) A coin is tossed once.
  • Outcomes: Head (H), Tail (T).
• (ii) A coin is tossed two times or two coins are tossed together once.
  • Outcomes: HH, HT, TH, TT.
• (iii) A coin is tossed three times.
  • Outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Some Basic Terms

• Trial:
  • A single action in an experiment is called a trial.
  • Examples:
    • Each toss of a coin is a trial.
    • Each throw of a die is a trial.
    • Each simultaneous toss of two coins is a trial.
• Equally Likely Outcomes:
  • Outcomes that have the same chance of happening are called equally likely outcomes.
  • Examples:
    • Tossing a coin heads or tails has an equal chance.
    • Throwing a die (each number 1 to 6 has an equal chance).
    • Drawing a card from a deck of 52 cards (each card has equal chance).
• Event:
  • Each outcome of an experiment is called an event.
• Discussion on Outcomes:
  • If a coin is tossed 100 times and heads appear 53 times, tails appear 47 times (100 - 53 = 47).
  • In the next toss, the chance of heads is based on past results (53/100), and tails is 47/100.
  • Favourable outcomes for heads = 53, for tails = 47.
• Example 2:Write all possible outcomes when:
  • (i) A die is thrown once.
    • Outcomes: 1, 2, 3, 4, 5, 6.
  • (ii) A die is thrown two times.
    • Outcomes: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6).

Probability

• Probability measures how likely an event is to happen in a random experiment.
• If there are n total outcomes and favourable outcomes for an event E, the probability of E, denoted P(E), is:
  • P(E) = Number of favourable outcomes / Total number of outcomes = m/n.
• Sure Event:
  • An event that will definitely happen is a sure event.
  • The probability of a sure event is always 1 (P(E) = 1).
  • Example: When a die is thrown, getting a number from 1 to 6 is a sure event.
  • Example 3:Can the probability of an event be 1.25? Give a reason.
    • Answer: No, the probability cannot be 1.25.
    • Reason: Probability is never more than 1.
• Impossible Event:
  • An event that cannot happen is an impossible event.
  • The probability of an impossible event is always 0 (P(E) = 0).
  • Example: When a die is thrown, getting a number greater than 6 is an impossible event.
  • Example 4:Can the probability of an event be -2? Give a reason.
    • Answer: No, the probability cannot be -2.
    • Reason: Probability is never less than 0.
• Probability Range:
  • The probability of any event E is always between 0 and 1 (0 ≤ P(E) ≤ 1).
• Example 5:A die is thrown once. Find the probability of getting:
  • (i) An even number.
    • Total outcomes = 6 (1, 2, 3, 4, 5, 6).
    • Favourable outcomes = 3 (2, 4, 6).
    • Probability = 3/6 = 1/2.
  • (ii) A multiple of 3.
    • Favourable outcomes = 2 (3, 6).
    • Probability = 2/6 = 1/3.
  • (iii) Not a multiple of 3.
    • Favourable outcomes = 4 (1, 2, 4, 5).
    • Probability = 4/6 = 2/3.
• Example 6:Two coins are tossed together. Find the probability of getting:
  • Total outcomes = 4 (HH, HT, TH, TT).
  • (i) No tail.
    • Favourable outcomes = 1 (HH).
    • Probability = 1/4.
  • (ii) Two tails.
    • Favourable outcomes = 1 (TT).
    • Probability = 1/4.
  • (iii) Exactly one tail.
    • Favourable outcomes = 2 (HT, TH).
    • Probability = 2/4 = 1/2.
• Example 7: A letter is chosen from the word 'TRIANGLE'. What is the probability that it is a vowel?
  • Total outcomes = 8 letters (T, R, I, A, N, G, L, E).
  • Favourable outcomes = 3 vowels (A, E, I).
  • Probability = 3/8.
• Example 8: A bag contains 4 red, 6 black, and 5 white balls. A ball is drawn at random. Find the probability that the ball drawn is:
  • Total outcomes = 4 + 6 + 5 = 15 balls.
  • (i) White.
    • Favourable outcomes = 5 (white balls).
    • Probability = 5/15 = 1/3.
  • (ii) Not black.
    • Favorable outcomes = 4 (red) + 5 (white) = 9.
    • Probability = 9/15 = 3/5.
  • (iii) Red or black.
    • Favorable outcomes = 4 (red) + 6 (black) = 10.
    • Probability = 10/15 = 2/3.