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Class 10 Maths Notes — Chapter 14: Probability

Classical (theoretical) probability of equally likely outcomes.

Detailed NCERT notes

  • Random experiment: outcome cannot be predicted with certainty.
  • Sample space S = set of all possible outcomes; event E ⊆ S.
  • For equally likely outcomes: P(E) = number of favourable outcomes / total number of outcomes.
  • 0 ≤ P(E) ≤ 1 for every event E. P(impossible) = 0, P(sure) = 1.
  • Complementary events: P(E) + P(not E) = 1, i.e. P(Ē) = 1 − P(E).
  • For a coin toss: S = {H, T}, 2 outcomes. Two coins: 4 outcomes. n coins: 2ⁿ outcomes.
  • For a die (6 faces): S = {1, 2, 3, 4, 5, 6}. Two dice: 36 outcomes (ordered pairs).
  • For a standard deck of 52 cards: 4 suits (spades, hearts, diamonds, clubs), each with 13 cards. Face cards = J, Q, K (12 total); Aces = 4; red cards = 26; black = 26.
  • Card-drawing 'without replacement' vs 'with replacement' matters — although at Class 10 level, single-draw is the usual setup.
  • For 'at least' questions, complement is often easier: P(at least one) = 1 − P(none).
  • Always write the total outcomes and favourable outcomes explicitly for full marks.

Formulas & key results

  • P(E) = favourable outcomes / total outcomes
  • 0 ≤ P(E) ≤ 1
  • P(E) + P(not E) = 1

Mind map

  • Sample space → event → equally likely
  • Complementary events
  • Coins, dice, cards contexts

Tricks & shortcuts

  • List sample space explicitly for two-dice/coin problems.
  • Use complement for 'at least one' style questions.

Common mistakes to avoid

  • Assuming outcomes are equally likely without checking.
  • Confusing 'at least' with 'exactly'.
  • Forgetting there are 12 face cards (or 16 if aces counted).

Competency-based questions & answers

  1. Q. Two dice are thrown. Find P(sum = 8).
    A. Favourable: (2,6), (3,5), (4,4), (5,3), (6,2) → 5. P = 5/36.
  2. Q. One card is drawn from a well-shuffled deck. Find P(it is a red face card).
    A. Red face cards = 6 (J, Q, K of hearts and diamonds). P = 6/52 = 3/26.
  3. Q. A bag has 3 red, 5 black, 4 white balls. One is drawn. Find P(not red).
    A. Total = 12; not red = 9. P = 9/12 = 3/4.