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Mathematics: Class 10 — Chapter 14: Probability

14.1 Probability — A Theoretical Approach

Key Concepts in Probability

  • Fair Coin / Unbiased Coin: A symmetrical coin with no reason to land on one side more often than the other.
  • Random Toss: The coin is allowed to fall freely without any bias or interference.
  • Equally Likely Outcomes: Outcomes that have the same possibility of occurring (e.g., head/tail for a coin, 1-6 for a die).
  • Experimental / Empirical Probability P(E) (Class IX definition):
    P(E) = Number of trials in which the event happened / Total number of trials
  • Theoretical Probability / Classical Probability P(E):
    P(E) = Number of outcomes favourable to E / Number of all possible outcomes of the experiment
    This definition assumes that all outcomes of the experiment are equally likely. It was given by Pierre Simon Laplace in 1795.
  • Elementary Event: An event having only one outcome of the experiment.
  • Complementary Events: If E is an event, then 'not E' is its complementary event, denoted as Ē.
    P(E) + P(Ē) = 1 or P(Ē) = 1 - P(E)
  • Impossible Event: An event that cannot happen. Its probability is 0.
  • Sure Event / Certain Event: An event that is certain to happen. Its probability is 1.
  • Probability Range: The probability of any event E always lies between 0 and 1, inclusive.
    0 ≤ P(E) ≤ 1
  • The sum of the probabilities of all the elementary events of an experiment is 1.

Examples

  • Example 1: Coin Toss
    • Possible outcomes: Head (H), Tail (T) = 2.
    • P(Head) = 1/2.
    • P(Tail) = 1/2.
  • Example 2: Drawing a Ball (1 red, 1 blue, 1 yellow)
    • Possible outcomes: Red, Blue, Yellow = 3.
    • P(Yellow) = 1/3. P(Red) = 1/3. P(Blue) = 1/3.
  • Example 3: Die Throw
    • Possible outcomes: 1, 2, 3, 4, 5, 6 = 6.
    • P(number > 4) = P(5 or 6) = 2/6 = 1/3.
    • P(number ≤ 4) = P(1, 2, 3, or 4) = 4/6 = 2/3.
  • Example 4: Drawing a Card (from a 52-card deck)
    • Total outcomes = 52.
    • P(be an ace) = 4/52 = 1/13.
    • P(not be an ace) = 48/52 = 12/13 (or 1 - 1/13).
  • Example 5: Tennis Match
    • P(Sangeeta wins) = 0.62.
    • P(Reshma wins) = 1 - 0.62 = 0.38 (complementary events).
  • Example 6: Birthdays (365 days, ignoring leap year)
    • P(different birthdays) = 364/365.
    • P(same birthday) = 1 - 364/365 = 1/365.
  • Example 7: Class Representative (25 girls, 15 boys, total 40)
    • P(girl) = 25/40 = 5/8.
    • P(boy) = 15/40 = 3/8.
  • Example 8: Marbles in a Box (3 blue, 2 white, 4 red, total 9)
    • P(white) = 2/9.
    • P(blue) = 3/9 = 1/3.
    • P(red) = 4/9.
  • Example 9: Two Coins Tossed
    • Possible outcomes: (H,H), (H,T), (T,H), (T,T) = 4.
    • P(at least one head) = 3/4.
  • Example 10: Musical Chair (Geometric Probability)
    • Music stops within 2 minutes (total range 2).
    • Music stops within first half-minute (favourable range 0.5).
    • P(stop within first half-minute) = 0.5 / 2 = 1/4.
  • Example 11: Helicopter Crash (Geometric Probability)
    • Total rectangular region area: 4.5 km × 9 km = 40.5 km².
    • Lake area: 2.5 km × 3 km = 7.5 km².
    • P(crashed in lake) = 7.5 / 40.5 = 5/27.
  • Example 12: Shirts with Defects (88 good, 8 minor, 4 major, total 100)
    • P(acceptable to Jimmy - only good) = 88/100 = 0.88.
    • P(acceptable to Sujatha - good or minor defects) = (88+8)/100 = 96/100 = 0.96.
  • Example 13: Two Dice Thrown (blue and grey)
    • Total outcomes: 6 × 6 = 36.
    • P(sum is 8) = 5/36 (outcomes: (2,6), (3,5), (4,4), (5,3), (6,2)).
    • P(sum is 13) = 0/36 = 0 (impossible event).
    • P(sum ≤ 12) = 36/36 = 1 (sure event).

Exercise 14.1

  • Complete statements on probability definitions.
  • Identify experiments with equally likely outcomes.
  • Explain why coin tossing is fair for team selection.
  • Identify invalid probability values.
  • Calculate P(not E) given P(E).
  • Calculate probabilities for drawing specific candies from a bag.
  • Calculate probability of same birthday given probability of different birthdays.
  • Calculate probabilities for drawing red/not red balls from a bag.
  • Calculate probabilities for drawing red/white/not green marbles from a box.
  • Calculate probabilities for coins falling out of a piggy bank.
  • Calculate probability of drawing a male fish.
  • Calculate probabilities for a spinning arrow landing on specific numbers.
  • Calculate probabilities for a die throw (prime, between 2 and 6, odd).
  • Calculate probabilities for drawing specific cards from a deck.
  • Calculate probabilities for drawing cards from a specific set of five cards (queen, then ace/queen after queen is removed).
  • Calculate probability of drawing a good pen from a mixed lot.
  • Calculate probabilities for drawing defective/non-defective bulbs (with replacement scenario).
  • Calculate probabilities for drawing discs with specific numbers (two-digit, perfect square, divisible by 5).
  • Calculate probabilities for a die with letters on faces.
  • Calculate geometric probability for a die landing inside a circle.
  • Calculate probabilities for Nuri buying/not buying a pen from a lot with defective pens.
  • Complete a probability table for the sum of two dice and evaluate a student's argument.
  • Calculate probability of losing a game involving tossing a coin 3 times.
  • Calculate probabilities for a die thrown twice (5 not up either time, 5 up at least once).
14.2 Summary