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Complete NCERT Solutions for CBSE Class 10 Maths Chapter 14 — Probability: theoretical probability with coins, dice, cards, and bags of balls.
Key Concepts (Quick Recap)
- P(E) = favourable outcomes / total outcomes; 0 ≤ P(E) ≤ 1.
- P(not E) = 1 − P(E).
- Standard deck: 52 cards, 26 red, 26 black, 12 face cards, 4 aces.
Exercise 14.1 — Theoretical Probability
Q1. A die is thrown once. Find the probability of getting (i) a prime number (ii) a number greater than 4 (iii) an odd number. Total outcomes = 6. (i) Primes {2, 3, 5} → 3/6 = 1/2. (ii) {5, 6} → 2/6 = 1/3. (iii) Odd {1, 3, 5} → 3/6 = 1/2.
Q2. Two coins are tossed simultaneously. Find the probability of getting (i) two heads (ii) at least one head (iii) no head. Sample space {HH, HT, TH, TT} → 4 outcomes. (i) {HH} → 1/4. (ii) {HH, HT, TH} → 3/4. (iii) {TT} → 1/4.
Q3. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting (i) a king (ii) a red card (iii) a face card (iv) the queen of hearts. (i) 4 kings → 4/52 = 1/13. (ii) 26 red → 26/52 = 1/2. (iii) 12 face cards → 12/52 = 3/13. (iv) 1 queen of hearts → 1/52.
Q4. A bag contains 3 red, 5 black and 2 white balls. A ball is drawn at random. Find the probability that it is (i) red (ii) not black (iii) white. Total = 10. (i) 3/10. (ii) not black = 1 − 5/10 = 1/2. (iii) 2/10 = 1/5.
Q5. A box contains 90 discs numbered 1 to 90. One disc is drawn at random. Find the probability that it bears (i) a two-digit number (ii) a perfect square (iii) a number divisible by 5. Total = 90. (i) Two-digit numbers are 10 to 90 → 81 numbers → 81/90 = 9/10. (ii) Perfect squares 1,4,9,16,25,36,49,64,81 → 9 → 9/90 = 1/10. (iii) Multiples of 5: 5,10,…,90 → 18 → 18/90 = 1/5.
Q6. A jar contains green, blue and white marbles. The probability of drawing a green marble is 1/4 and a blue marble is 1/3. If the jar has 24 marbles, how many are white? P(white) = 1 − 1/4 − 1/3 = (12 − 3 − 4)/12 = 5/12. White marbles = 5/12 × 24 = 10.
Common Mistakes
- Listing the sample space for two coins as 3 outcomes — it is 4 (HT and TH are different).
- Counting two-digit numbers from 1 instead of 10 (there are 81, not 90).
- Forgetting there are 12 face cards (J, Q, K × 4 suits), not 16.
Exam Tips
- Write the total number of outcomes first, then count the favourable ones.
- Use P(not E) = 1 − P(E) when "not" or "at least" makes direct counting harder.
- For card problems, memorise the deck breakdown (26 red, 12 face, 4 aces, 4 of each rank).