Mathematics: Class 9 — Chapter 7: The Mathematics of Maybe: Introduction to Probability

• Randomness: A situation or action where you cannot predict exactly what will happen, even if all possible outcomes are known.
• Random Experiment (or Trial): An action that can be repeated where the result might be different each time, and the exact outcome cannot be known in advance (e.g., tossing a coin, rolling a die).

Probability is measured on a scale from 0 to 1 (or 0% to 100%):

• 0 (Impossible): The event cannot happen.
• 0.25 (Less likely): Low chance of occurrence.
• 0.5 (Equally likely / Even chance): 50/50 chance (e.g., flipping a fair coin).
• 0.75 (More likely): High chance of occurrence.
• 1 (Certain): The event is guaranteed to happen.

• Based on actual data collected from trials or experiments.
• Formula:
  Experimental Probability = (Number of times the event occurred) / (Total number of trials)
• Relative Frequency: The ratio of the number of times an event occurs to the total number of trials.

• Calculated under ideal, perfectly fair situations where all outcomes are assumed to be equally likely. No physical experiment is required.
• Formula:
  Theoretical Probability P(E) = (Number of favourable outcomes) / (Total number of possible outcomes)
• Law of Large Numbers: As the number of trials in an experiment increases, the experimental probability tends to get closer to the theoretical probability.
• Gambler's Fallacy: The mistaken belief that past random trials affect future independent trials (e.g., thinking tails is "due" after rolling heads 6 times). Coins and dice have no memory.
• Fair and Unbiased: An object (like a coin or die) is unbiased if it is symmetrical and has an equal chance of landing on any of its faces.

• Population: The entire group under study (e.g., all 1500 students in a school).
• Sample: A smaller, representative subset of the population used to gather evidence (e.g., 50 students from one class).
• Sampling: The process of selecting a representative group to make predictions about the larger population.

• Sample Space (S): The set of all possible outcomes of a random experiment.
  • Must include every possible outcome.
  • No outcome should be listed more than once.
• Sample Size (n(S)): The total number of elements in the sample space.
• Examples:
  • Tossing a coin: S = {H, T}, n(S) = 2
  • Rolling a die: S = {1, 2, 3, 4, 5, 6}, n(S) = 6
  • Tossing two coins: S = {HH, HT, TH, TT}, n(S) = 4

• Event (E): Any single possible result or combination of results; a subset of the sample space.
• Examples:
  • Rolling a die and getting a number > 4: E = {5, 6}
  • Tossing two coins and getting at least one head: E = {HH, HT, TH}