Probability: Class 10 Important Questions with Answers

20 MCQs (1 mark each)

1. P(getting a 7 on a die) is
   (a) $\frac{1}{6}$ (b) 0 (c) 1 (d) $\frac{7}{6}$
2. If P(E) = 0.35, P(not E) is
   (a) 0.65 (b) 0.35 (c) 1.35 (d) 0
3. Two coins: P(no head)
   (a) $\frac{1}{4}$ (b) $\frac{1}{2}$ (c) $\frac{3}{4}$ (d) 0
4. A die: P(number > 4)
   (a) $\frac{1}{3}$ (b) $\frac{1}{2}$ (c) $\frac{2}{3}$ (d) $\frac{1}{6}$
5. Cards: P(a king)
   (a) $\frac{1}{52}$ (b) $\frac{1}{13}$ (c) $\frac{1}{4}$ (d) $\frac{4}{13}$
6. Two dice: P(sum 7)
   (a) $\frac{1}{6}$ (b) $\frac{7}{36}$ (c) $\frac{5}{36}$ (d) $\frac{1}{12}$
7. Two dice: P(doublet)
   (a) $\frac{1}{6}$ (b) $\frac{1}{36}$ (c) $\frac{1}{3}$ (d) $\frac{5}{6}$
8. Numbers 1–25: P(prime)
   (a) $\frac{8}{25}$ (b) $\frac{9}{25}$ (c) $\frac{2}{5}$ (d) $\frac{1}{5}$
9. Which can be a probability?
   (a) 1.01 (b) −0.2 (c) $\frac{5}{4}$ (d) 0.99
10. Three coins: P(exactly 2 heads)
    (a) $\frac{1}{8}$ (b) $\frac{3}{8}$ (c) $\frac{1}{2}$ (d) $\frac{7}{8}$
11. A letter from "PROBABILITY": P(B)
    (a) $\frac{1}{11}$ (b) $\frac{2}{11}$ (c) $\frac{3}{11}$ (d) $\frac{1}{5}$
12. Cards: P(black face card)
    (a) $\frac{3}{26}$ (b) $\frac{3}{13}$ (c) $\frac{1}{26}$ (d) $\frac{6}{13}$
13. Discs 1–50: P(multiple of 6)
    (a) $\frac{4}{25}$ (b) $\frac{8}{50}$ (c) both (a) and (b) (d) $\frac{1}{6}$
14. Two dice: P(product 6)
    (a) $\frac{1}{9}$ (b) $\frac{1}{6}$ (c) $\frac{1}{12}$ (d) $\frac{2}{9}$
15. A bag has 5 red and x blue balls; P(red) = $\frac{1}{3}$. x is
    (a) 5 (b) 10 (c) 15 (d) 3
16. P(sure event) is
    (a) 0 (b) 1 (c) $\frac{1}{2}$ (d) any value
17. A die: P(even prime)
    (a) $\frac{1}{2}$ (b) $\frac{1}{3}$ (c) $\frac{1}{6}$ (d) 0
18. A non-leap year: P(53 Mondays)
    (a) $\frac{1}{7}$ (b) $\frac{2}{7}$ (c) $\frac{53}{365}$ (d) $\frac{1}{52}$
19. Two dice: P(sum is a multiple of 3)
    (a) $\frac{1}{3}$ (b) $\frac{1}{4}$ (c) $\frac{5}{12}$ (d) $\frac{1}{6}$
20. Cards: P(neither a heart nor a king)
    (a) $\frac{9}{13}$ (b) $\frac{3}{4}$ (c) $\frac{4}{13}$ (d) $\frac{1}{2}$

Answer key

1 (b) · 2 (a) · 3 (a) · 4 (a) · 5 (b) · 6 (a) · 7 (a) · 8 (b) · 9 (d) · 10 (b) · 11 (b) · 12 (a) · 13 (c) · 14 (a) · 15 (b) · 16 (b) · 17 (c) · 18 (a) · 19 (a) · 20 (a)

Hints: Q8 2, 3, 5, 7, 11, 13, 17, 19, 23. Q13 6, 12, …, 48: 8. Q14 (1,6), (6,1), (2,3), (3,2). Q15 $\frac{5}{5+x}=\frac{1}{3}$. Q19 sums 3, 6, 9, 12: 2 + 5 + 4 + 1 = 12. Q20 hearts or kings = 13 + 3 = 16 ⇒ $\frac{36}{52}$.

Assertion–Reason (10)

(a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true.

1. A: P(getting a number less than 7 on a die) = 1. R: It is a sure event.
2. A: Tossing two coins has 3 equally likely outcomes. R: The number of heads can be 0, 1 or 2.
3. A: P(E) = 1.2 is not possible. R: 0 ≤ P(E) ≤ 1.
4. A: P(a red queen) = $\frac{1}{26}$. R: There are 2 red queens in a pack.
5. A: The sums 2 to 12 on two dice are equally likely. R: There are 11 possible sums.
6. A: If P(E) = 0.2, P(not E) = 0.8. R: P(E) + P(not E) = 1.
7. A: A leap year has 53 Sundays with probability $\frac{2}{7}$. R: A leap year has 52 weeks and 2 days.
8. A: The probability of an impossible event is 0. R: An impossible event has no favourable outcomes.
9. A: P(a prime on a die) = $\frac{1}{2}$. R: 1 is a prime number.
10. A: Cards 1 to 30: P(a multiple of 4) = $\frac{7}{30}$. R: The multiples of 4 up to 30 are 4, 8, …, 28.

Key: 1 (a) · 2 (d) · 3 (a) · 4 (a) · 5 (d) · 6 (a) · 7 (a) · 8 (a) · 9 (c) · 10 (a)

Q2: there are 4 outcomes (HT ≠ TH) ⇒ A false, R true. Q9: primes 2, 3, 5 ⇒ A true; 1 is not prime ⇒ R false.

Very short answer (1 mark each)

1. Two dice: how many outcomes? Ans. 36
2. A die: P(a multiple of 3). Ans. $\frac{1}{3}$
3. Cards: P(an ace of spades). Ans. $\frac{1}{52}$
4. If P(not E) = $\frac{3}{8}$, find P(E). Ans. $\frac{5}{8}$
5. Discs 1–100: P(a two-digit number). Ans. $\frac{90}{100}=\frac{9}{10}$
6. Three coins: P(all tails). Ans. $\frac{1}{8}$
7. A bag: 3 white, 2 black. P(black). Ans. $\frac{2}{5}$
8. Two dice: P(sum 13). Ans. 0

Short answer practice (2 marks each)

1. A die is thrown twice. Find the probability that the sum of the numbers is (i) 8 (ii) at most 4.

2. From a pack of 52 cards, the black kings and red queens are removed. One card is drawn from the rest. Find P(a face card).

Short answer practice (3 marks each)

3. A box contains 12 balls of which x are black. (i) Find P(black). (ii) If 6 more black balls are put in, the probability of black becomes double. Find x.

4. Cards numbered 2 to 101 are placed in a box. A card is drawn at random. Find the probability that it has (i) an even number (ii) a perfect square (iii) a number divisible by 5.

5. Two dice are thrown. Find the probability that (i) the numbers are both even (ii) the product is odd (iii) 6 appears on at least one die.

6. A game: a coin is tossed three times; if one or two heads come, Ramesh wins ₹10, otherwise he pays ₹15. Find the probability that he wins and that he loses.