Probability Chapter Notes | Year 6 Mathematics IGCSE (Cambridge) PDF Download

Getting Started

• Probability concepts:
  • Probability describes how likely an event is to occur.
  • Likelihood scale: Ranges from impossible (0% chance) to certain (100% chance), with unlikely, even chance, and likely in between.
• Card experiment:
  • Randomly selecting a card from a set involves assessing probabilities based on the number of each type of card.
  • Equally likely outcomes: Each card has the same chance of being selected if the set is uniform.
  • Comparative likelihood: Probability of selecting one card type (e.g., 3) versus another (e.g., 5) depends on their frequencies.
• Coin flip experiment:
  • Flipping a coin has two equally likely outcomes: heads or tails (each with a 50% chance).
  • Misconception: Previous flips do not influence future flips; each flip is independent, so the probability remains 50% for heads or tails.

Describing and Predicting Likelihood

• Objectives:
  • Describe the chance of outcomes using fractions and percentages.
  • Understand mutually exclusive events.
  • Use likelihood to predict outcomes.
  • Conduct chance experiments and describe results.
• Probability definition:
  • Probability quantifies how likely an event is to happen, aiding in predicting future outcomes.
  • Used in real-world scenarios (e.g., deciding whether to play a claw machine game based on the likelihood of winning a toy).
• Key terms:
  • Event: A specific outcome or set of outcomes (e.g., picking a red ball).
  • Outcome: A single result of an experiment (e.g., a specific card drawn).
  • Equally likely outcomes: Each outcome has the same probability (e.g., heads or tails in a coin flip).
  • Mutually exclusive events: Events that cannot occur simultaneously (e.g., spinning a 5 and spinning a number less than 4).
  • Probability: A measure of likelihood, expressed as a fraction, decimal, or percentage.
  • Probability experiment: A repeatable process with observable outcomes (e.g., flipping a coin multiple times).
• Describing probability:
  • Fractions: Probability = (Number of favorable outcomes) ÷ (Total number of outcomes).
    • Example: In a bag with 4 balls (2 red, 1 yellow, 1 green), the probability of picking a red ball is 2/4 = 1/2.
  • Percentages: Convert fraction to percentage (e.g., 1/2 = 50%).
    • Example: Weather forecast shows a 23% chance of rain, meaning 23 out of 100 times it might rain under similar conditions.
• Mutually exclusive events:
  • Events are mutually exclusive if they cannot happen at the same time.
    • Example: On a spinner with numbers 1–5, spinning a 5 and spinning a number less than 4 are mutually exclusive (5 is not less than 4).
    • Counterexample: Spinning a 5 and spinning a number greater than 2 are not mutually exclusive (5 is greater than 2).
  • Venn diagrams can illustrate mutually exclusive events:
    • For tickets numbered 1–30, odd numbers and multiples of 10 are mutually exclusive for a big prize (no number is both odd and a multiple of 10).
• Experimental probability:
  • Based on observed results from a probability experiment.
  • Formula: Experimental probability = (Number of times an event occurs) ÷ (Total number of trials).
    • Example: In 20 coin flips, if tails occurs 8 times, the experimental probability of tails is 8/20 = 2/5 = 40%.
  • Larger numbers of trials provide more reliable estimates of true probability.
• Predicting outcomes:
  • Use known probabilities to predict future results.
    • Example: A spinner with 4 red, 3 blue, and 1 yellow section (8 total sections).
      • Probability of red = 4/8 = 1/2 = 50%.
      • Probability of blue = 3/8 = 37.5%.
      • Probability of yellow = 1/8 = 12.5%.
      • For 8 spins, expect approximately 4 red, 3 blue, 1 yellow.
  • Scale predictions for more trials:
    • For 16 spins: 16 × (4/8) = 8 red, 16 × (3/8) = 6 blue, 16 × (1/8) = 2 yellow.
    • For 40 spins: 40 × (4/8) = 20 red, 40 × (3/8) = 15 blue, 40 × (1/8) = 5 yellow.
    • For 200 spins: 200 × (4/8) = 100 red, 200 × (3/8) = 75 blue, 200 × (1/8) = 25 yellow.
• Dice experiments:
  • Two six-sided dice (red and blue) have 36 possible outcomes (6 × 6).
  • Events:
    • Double: Both dice show the same number (e.g., (1,1), (2,2), ..., (6,6); 6/36 = 1/6 ≈ 16.67%).
    • Sum is even: Sum of dice is divisible by 2 (18/36 = 1/2 = 50%).
    • Blue die > red die: Blue die’s number is higher (15/36 ≈ 41.67%).
    • Red die is 6: Red die shows 6 (6/36 = 1/6 ≈ 16.67%).
  • Conditional probability:
    • If the red die is 6, probabilities adjust:
      • Double: Only (6,6) is possible (1/6 ≈ 16.67%).
      • Sum is even: Sum = 6 + blue (7, 8, 9, 10, 11, 12; 3 even sums: 8, 10, 12; 3/6 = 50%).
      • Blue > red: Blue must be > 6 (impossible; 0%).
      • Red is 6: Already true (100%).