NCERT Class 10 Maths – Exercise 14.1 Solutions

NCERT Class 10 Maths

Chapter 14 – Probability | Exercise 14.1

(Rationalized Syllabus 2025-26)

💡 Probability Formulas

Probability P(E): $\frac{\text{No. of favorable outcomes}}{\text{Total no. of possible outcomes}}$
$P(E) + P(\text{not } E) = 1$
$0 \le P(E) \le 1$

Complete the following statements:

(i) Probability of an event E + Probability of the event ‘not E’ = 1.

(ii) The probability of an event that cannot happen is 0. Such an event is called an impossible event.

(iii) The probability of an event that is certain to happen is 1. Such an event is called a sure event.

(iv) The sum of the probabilities of all the elementary events of an experiment is 1.

(v) The probability of an event is greater than or equal to 0 and less than or equal to 1.

Which of the following experiments have equally likely outcomes? Explain.

(i) A driver attempts to start a car. The car starts or does not start.

❌ Not equally likely

Reason: It depends on the condition of the car. A working car starts usually; a broken one doesn’t.

(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.

❌ Not equally likely

Reason: It depends on the player’s ability.

(iii) A trial is made to answer a true-false question. The answer is right or wrong.

✔️ Equally likely

Reason: There are only two possibilities and both are fair chances.

(iv) A baby is born. It is a boy or a girl.

✔️ Equally likely

Reason: Biologically, the chance is 50-50.

Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a football game?

When a coin is tossed, there are only two possible outcomes: Head or Tail. These outcomes are equally likely. The result of an individual toss is completely unpredictable. Therefore, it is considered a fair method.

Which of the following cannot be the probability of an event? (A) 2/3 (B) -1.5 (C) 15% (D) 0.7

Probability is always between 0 and 1 inclusive. It cannot be negative.

✔️ (B) -1.5

If P(E) = 0.05, what is the probability of ‘not E’?

$$P(\text{not } E) = 1 – P(E) = 1 – 0.05 = 0.95$$

✔️ 0.95

A bag contains lemon flavoured candies only. Malini takes out one candy without looking into the bag. What is the probability that she takes out (i) an orange flavoured candy? (ii) a lemon flavoured candy?

(i) Orange flavoured candy: The bag contains only lemon candies. Drawing an orange candy is an impossible event.

$$P(\text{Orange}) = 0$$

(ii) Lemon flavoured candy: Since all candies are lemon, this is a sure event.

$$P(\text{Lemon}) = 1$$

It is given that in a group of 3 students, the probability of 2 students not having the same birthday is 0.992. What is the probability that the 2 students have the same birthday?

Let $E$ be the event of having the same birthday.

$P(\text{not } E) = 0.992$.

$$P(E) = 1 – P(\text{not } E) = 1 – 0.992 = 0.008$$

✔️ 0.008

A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is (i) red? (ii) not red?

Total balls = 3 (Red) + 5 (Black) = 8.

(i) P(Red) = $\frac{3}{8}$.

(ii) P(Not Red) = P(Black) = $\frac{5}{8}$ (or $1 – 3/8$).

✔️ (i) 3/8 (ii) 5/8

A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out of the box at random. What is the probability that the marble taken out will be (i) red? (ii) white? (iii) not green?

Total marbles = $5 + 8 + 4 = 17$.

(i) Red

$$P(\text{Red}) = \frac{5}{17}$$

(ii) White

$$P(\text{White}) = \frac{8}{17}$$

(iii) Not Green

Not green means Red or White. Count = $5 + 8 = 13$.

$$P(\text{Not Green}) = \frac{13}{17}$$

Q10

A piggy bank contains hundred 50p coins, fifty ₹ 1 coins, twenty ₹ 2 coins and ten ₹ 5 coins. If it is equally likely that one of the coins will fall out… what is the probability that the coin (i) will be a 50p coin? (ii) will not be a ₹ 5 coin?

Total coins = 100 (50p) + 50 (₹1) + 20 (₹2) + 10 (₹5) = 180.

(i) 50p coin

$$P(50p) = \frac{100}{180} = \frac{5}{9}$$

(ii) Not ₹ 5 coin

Coins that are not ₹5 = $180 – 10 = 170$.

$$P(\text{Not ₹5}) = \frac{170}{180} = \frac{17}{18}$$

Q11

Gopi buys a fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing 5 male fish and 8 female fish. What is the probability that the fish taken out is a male fish?

Total fish = 5 (Male) + 8 (Female) = 13.

Favorable outcomes (Male) = 5.

$$P(\text{Male}) = \frac{5}{13}$$

✔️ 5/13

Q12

A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers 1, 2, 3, 4, 5, 6, 7, 8… What is the probability that it will point at (i) 8? (ii) an odd number? (iii) a number greater than 2? (iv) a number less than 9?

Total outcomes = 8.

(i) 8: 1 favorable outcome. $P = 1/8$.

(ii) Odd number: {1, 3, 5, 7} = 4 outcomes. $P = 4/8 = 1/2$.

(iii) Greater than 2: {3, 4, 5, 6, 7, 8} = 6 outcomes. $P = 6/8 = 3/4$.

(iv) Less than 9: {1…8} = 8 outcomes. $P = 8/8 = 1$.

Q13

A die is thrown once. Find the probability of getting (i) a prime number; (ii) a number lying between 2 and 6; (iii) an odd number.

Total outcomes = {1, 2, 3, 4, 5, 6} = 6.

(i) Prime: {2, 3, 5} = 3 outcomes. $P = 3/6 = 1/2$.

(ii) Between 2 and 6: {3, 4, 5} = 3 outcomes. $P = 3/6 = 1/2$.

(iii) Odd: {1, 3, 5} = 3 outcomes. $P = 3/6 = 1/2$.

Q14

One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting: (i) a king of red colour (ii) a face card (iii) a red face card (iv) the jack of hearts (v) a spade (vi) the queen of diamonds.

Total cards = 52.

(i) Red King: (Hearts, Diamonds) = 2. $P = 2/52 = 1/26$.

(ii) Face Card: (J, Q, K in 4 suits) = 12. $P = 12/52 = 3/13$.

(iii) Red Face Card: 6 cards. $P = 6/52 = 3/26$.

(iv) Jack of Hearts: 1 card. $P = 1/52$.

(v) Spade: 13 cards. $P = 13/52 = 1/4$.

(vi) Queen of Diamonds: 1 card. $P = 1/52$.

Q15

Five cards—the ten, jack, queen, king and ace of diamonds, are well-shuffled with their face downwards. One card is then picked up at random. (i) What is the probability that the card is the queen? (ii) If the queen is drawn and put aside, what is the probability that the second card picked up is (a) an ace? (b) a queen?

(i) Total cards = 5. Queens = 1.

$$P(\text{Queen}) = \frac{1}{5}$$

(ii) Queen put aside. Total cards remaining = 4 (10, J, K, A).

(a) Ace: 1 ace in 4 cards. $P(\text{Ace}) = 1/4$.

(b) Queen: 0 queens left. $P(\text{Queen}) = 0$.

Q16

12 defective pens are accidentally mixed with 132 good ones… One pen is taken out at random… Determine the probability that the pen taken out is a good one.

Total pens = 12 (Defective) + 132 (Good) = 144.

Favorable outcomes (Good) = 132.

$$P(\text{Good}) = \frac{132}{144} = \frac{11}{12}$$

✔️ 11/12

Q17

(i) A lot of 20 bulbs contain 4 defective ones. One bulb is drawn at random from the lot. What is the probability that this bulb is defective? (ii) Suppose the bulb drawn in (i) is not defective and is not replaced. Now one bulb is drawn at random from the rest. What is the probability that this bulb is not defective?

(i) Total = 20. Defective = 4.

$$P(\text{Defective}) = \frac{4}{20} = \frac{1}{5}$$

(ii) Bulb drawn was good and not replaced.

New Total = 19. Total Good = 16 – 1 = 15.

$$P(\text{Not Defective}) = \frac{15}{19}$$

Q18

A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears (i) a two-digit number (ii) a perfect square number (iii) a number divisible by 5.

Total outcomes = 90.

(i) Two-digit numbers (10 to 90): Count = $90 – 9 = 81$.

$$P = \frac{81}{90} = \frac{9}{10}$$

(ii) Perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81): Count = 9.

$$P = \frac{9}{90} = \frac{1}{10}$$

(iii) Divisible by 5 (5, 10…90): Count = 18.

$$P = \frac{18}{90} = \frac{1}{5}$$

Q19

A child has a die whose six faces show the letters: A, B, C, D, E, A. The die is thrown once. What is the probability of getting (i) A? (ii) D?

Total faces = 6.

(i) Getting A: ‘A’ appears 2 times. $P(A) = 2/6 = 1/3$.

(ii) Getting D: ‘D’ appears 1 time. $P(D) = 1/6$.

Q20

Suppose you drop a die at random on the rectangular region… What is the probability that it will land inside the circle with diameter 1m? (Rect dimensions 3m x 2m)

Area of Rectangle = $3 \times 2 = 6 \text{ m}^2$.

Diameter of Circle = 1 m $\Rightarrow$ Radius = 0.5 m.

Area of Circle = $\pi r^2 = \pi (0.5)^2 = 0.25\pi = \pi/4 \text{ m}^2$.

$$P = \frac{\text{Area of Circle}}{\text{Area of Rectangle}} = \frac{\pi/4}{6} = \frac{\pi}{24}$$

✔️ $\pi/24$

Q21

A lot consists of 144 ball pens of which 20 are defective and the others are good. Nuri will buy a pen if it is good… The shopkeeper draws one pen at random… What is the probability that (i) She will buy it? (ii) She will not buy it?

Total pens = 144. Defective = 20. Good = $144 – 20 = 124$.

(i) She buys it (Good pen):

$$P(\text{Good}) = \frac{124}{144} = \frac{31}{36}$$

(ii) She will not buy it (Defective):

$$P(\text{Defective}) = \frac{20}{144} = \frac{5}{36}$$

Q22

Two dice are thrown at the same time. Complete the table for the probability of the sum of two numbers.

Total outcomes = $6 \times 6 = 36$.

Sum 2 (1,1): 1/36
Sum 3 (1,2; 2,1): 2/36
Sum 4 (1,3; 2,2; 3,1): 3/36
Sum 5: 4/36
Sum 6: 5/36
Sum 7: 6/36
Sum 8: 5/36
Sum 9: 4/36
Sum 10: 3/36
Sum 11: 2/36
Sum 12: 1/36

Q23

A game consists of tossing a one rupee coin 3 times and noting its outcome each time. Hanif wins if all the tosses give the same result i.e., three heads or three tails, and loses otherwise. Calculate the probability that Hanif will lose the game.

Total outcomes (8): {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT}.

Winning outcomes (Same result): {HHH, TTT} = 2.

Losing outcomes = $8 – 2 = 6$.

$$P(\text{Lose}) = \frac{6}{8} = \frac{3}{4}$$

✔️ 3/4

Q24

A die is thrown twice. What is the probability that (i) 5 will not come up either time? (ii) 5 will come up at least once?

Total outcomes = 36.

Outcomes with 5: {(1,5), (2,5), (3,5), (4,5), (5,5), (6,5), (5,1), (5,2), (5,3), (5,4), (5,6)}. Total = 11.

(i) 5 will not come up: $36 – 11 = 25$. $P = 25/36$.

(ii) 5 will come up at least once: 11 outcomes. $P = 11/36$.

✔️ (i) 25/36 (ii) 11/36

Q25

Which of the following arguments are correct and which are not correct? Give reasons.
(i) If two coins are tossed, there are three possible outcomes—two heads, two tails or one of each. Therefore, probability of each is 1/3.
(ii) If a die is thrown, probability of odd number is 1/2.

(i) Argument Analysis

Incorrect. The outcomes are {HH, HT, TH, TT}. “One of each” includes HT and TH, so its probability is 2/4 = 1/2, not 1/3.

❌ Incorrect

(ii) Argument Analysis

Correct. Outcomes are {1, 2, 3, 4, 5, 6}. Odd are {1, 3, 5} (3 outcomes). P(Odd) = 3/6 = 1/2.

✔️ Correct

🎉 Exercise 14.1 Completed | Chapter 14 Finished!