Probability helps students understand how likely an event is to occur and builds logical thinking for real-life situations involving chance and uncertainty. Chapter 15: Probability focuses on basic probability concepts, equally likely outcomes, finding the probability of events, and solving practical problems using simple calculations.
These NCERT Solutions for Class 10 Maths Chapter 14 cover all important CBSE board-level questions asked between 2019 and 2025. Each solution is explained step by step in clear and simple language to help students strengthen fundamentals and score confidently in the board exams.
Q.
Complete the following statements:
(i) Probability of an event E + Probability of the event ‘not E’ =_____.
(ii) The probability of an event that cannot happen is_____. Such an event is called _____.
(iii) The probability of an event that is certain to happen is_____. Such an event is called______.
(iv) The sum of the probabilities of all the elementary events of an experiment is _____.
(v) The probability of an event is greater than or equal to _____ and less than or equal to_____.
Q.
(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?
Q.
A box contains 12 balls out of which x are black. If one ball is drawn at random from the box, what is the probability that it will be a black ball? If 6 more black balls are put in the box, the probability of drawing a black ball is now double of what it was before. Find x.
Q.
A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, determine the number of blue balls in the bag.
Q.
Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday). Each is equally likely to visit the shop on any day as on another day. What is the probability that both will visit the shop on (i) the same day? (ii) consecutive days? (iii) different days?
Q.
Which of the following arguments are correct and which are not correct? Give reasons for your answer.
(i) If two coins are tossed simultaneously there are three possible outcomes — two heads, two tails or one of each. Therefore, for each of these outcomes, the probability is 1/3.
(ii) If a die is thrown, there are two possible outcomes — an odd number or an even number. Therefore, the probability of getting an odd number is 1/2.
Q.
Two dice, one blue and one grey, are thrown at the same time.
(i) Complete the following table:
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Event:
‘Sum of two dice’
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11
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12
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Probability
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(ii) A student argues that there are 11 possible outcomes 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12. Therefore, each of them has a probability 1/11. Do you agree with this argument? Justify your answer.
Q.
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, but will not buy if it is defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that
(i) She will buy it?
(ii) She will not buy it?
Q.
A child has a die whose six faces show the letters as given below:
The die is thrown once. What is the probability of getting (i) A? (ii) D?
Q.
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.
Q.
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.
(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.
(iii) A trial is made to answer a true-false question. The answer is right or wrong.
(iv) A baby is born. It is a boy or a girl.
Q.
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 (see the following figure). What is the probability that the fish taken out is a male fish?
Q.
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?
Q.
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?
Q.
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?
Q.
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?
Q.
If P(E) = 0.05, what is the probability of ‘not E’?
Q.
Which of the following cannot be the probability of an event?
(A) 2/3
(B) –1.5
(C) 15%
(D) 0.7
Q.
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?
Q.
A jar contains 24 marbles, some are green and others are blue. If a marble is drawn at random from the jar, the probability that it is green is 2/3⋅ Find the number of blue marbles in the jar.
NCERT Solutions for Class 10 Maths Chapter 14 – Probability
Q1.
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?
Ans:
Let E = event that “2 students do not have the same birthday”.
Given, P(E) = 0.992Required probability = P(not E) = 1 − P(E)
= 1 − 0.992
= 0.008So, the probability that 2 students have the same birthday is 0.008.
Q2.
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.
(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.
(iii) A trial is made to answer a true-false question. The answer is right or wrong.
(iv) A baby is born. It is a boy or a girl.
Ans:
Equally likely outcomes mean each outcome has the same chance of occurring.
(i) Not equally likely (car starting depends on many conditions).
(ii) Not equally likely (depends on player skill, situation, etc.).
(iii) Equally likely (if a student answers randomly, right/wrong chances are equal).
(iv) Equally likely (boy/girl outcomes are treated as equally likely at this level).
Hence, experiments (iii) and (iv) have equally likely outcomes.
Q3.
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?
Ans:
Total fish = 5 + 8 = 13
Favourable outcomes (male fish) = 5Probability of selecting a male fish = 5/13
= 5/13
Q4.
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
Ans:
Sample space S = {1, 2, 3, 4, 5, 6} so total outcomes = 6
(i) Prime numbers on a die: {2, 3, 5} → favourable = 3
P(prime) = 3/6 = 1/2
(ii) Numbers lying between 2 and 6 (i.e., 3,4,5): {3, 4, 5} → favourable = 3
P(between 2 and 6) = 3/6 = 1/2
(iii) Odd numbers: {1, 3, 5} → favourable = 3
P(odd) = 3/6 = 1/2
Q5.
(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?
Ans:
(i) Total bulbs = 20, defective bulbs = 4
P(defective) = 4/20 = 1/5(ii) First bulb drawn is not defective and not replaced.
So remaining bulbs = 19
Initially good bulbs = 20 − 4 = 16
Since one good bulb is removed, remaining good bulbs = 15P(not defective) = 15/19
= 15/19
Q6.
A child has a die whose six faces show the letters: A, B, C, D, E, F. The die is thrown once.
What is the probability of getting (i) A? (ii) D?
Ans:
Total outcomes = 6 (A, B, C, D, E, F)
(i) Favourable outcomes for A = 1
P(A) = 1/6 = 1/6
(ii) Favourable outcomes for D = 1
P(D) = 1/6 = 1/6
Q7.
A jar contains 24 marbles, some are green and others are blue. If a marble is drawn at random from the jar,
the probability that it is green is 2/3. Find the number of blue marbles in the jar.
Ans:
Let number of green marbles = G
Total marbles = 24
Given, P(green) = G/24 = 2/3G = 24 × (2/3) = 16
Blue marbles = 24 − 16 = 8So, the number of blue marbles is 8.
Q8.
A box contains 12 balls out of which x are black. If one ball is drawn at random from the box, what is the probability that it will be a black ball?
If 6 more black balls are put in the box, the probability of drawing a black ball is now double of what it was before. Find x.
Ans:
Initially: total balls = 12, black balls = x
P1 = x/12After adding 6 black balls: black balls = x + 6, total balls = 18
P2 = (x + 6)/18Given, P2 = 2 × P1
(x + 6)/18 = 2(x/12)
(x + 6)/18 = x/6
Cross multiply:
6(x + 6) = 18x
6x + 36 = 18x
36 = 12x
x = 3
So, number of black balls initially = 3.
Q9.
A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball,
determine the number of blue balls in the bag.
Ans:
Let number of blue balls = x
Total balls = 5 + xP(blue) = x/(5 + x)
P(red) = 5/(5 + x)Given, P(blue) = 2 × P(red)
x/(5 + x) = 2 × 5/(5 + x)
x = 10
So, number of blue balls = 10.
Q10.
Two customers Shyam and Ekta are visiting a particular shop in the same week (Tuesday to Saturday).
Each is equally likely to visit the shop on any day as on another day.
What is the probability that both will visit the shop on (i) the same day? (ii) consecutive days? (iii) different days?
Ans:
Days = Tuesday to Saturday = 5 days
Total possible pairs = 5 × 5 =
25(i) Same day: (T,T), (W,W), (Th,Th), (F,F), (S,S) → 5 outcomes
P(same day) = 5/25 =
1/5(ii) Consecutive days (order matters):
(Tue,Wed), (Wed,Tue), (Wed,Thu), (Thu,Wed), (Thu,Fri), (Fri,Thu), (Fri,Sat), (Sat,Fri)
Total = 8 outcomes
P(consecutive) = 8/25 =
8/25
(iii) Different days = 1 − P(same day)
= 1 − 1/5 = 4/5
Q11.
Which of the following arguments are correct and which are not correct? Give reasons.
(i) If two coins are tossed simultaneously there are three possible outcomes — two heads, two tails or one of each. Therefore, for each of these outcomes, the probability is 1/3.
(ii) If a die is thrown, there are two possible outcomes — an odd number or an even number. Therefore, the probability of getting an odd number is 1/2.
Ans:
(i) Not correct. Two coins have sample space {HH, HT, TH, TT} (4 outcomes).
“One of each” (HT, TH) has 2 outcomes, so probability is 2/4 = 1/2, not 1/3.(ii) Correct. For a fair die, odd outcomes = {1,3,5} (3) and even outcomes = {2,4,6} (3).
So, P(odd) = 3/6 = 1/2.
Q12.
Two dice, one blue and one grey, are thrown at the same time.
(i) Complete the probability table for the sum of two dice.
(ii) A student argues that there are 11 possible outcomes 2 to 12, so each has probability 1/11. Do you agree? Justify.
Ans:
Total outcomes when two dice are thrown = 6 × 6 = 36
(i) Probabilities of sums:
Sum 2: 1/36
Sum 3: 2/36 = 1/18
Sum 4: 3/36 = 1/12
Sum 5: 4/36 = 1/9
Sum 6: 5/36
Sum 7: 6/36 = 1/6
Sum 8: 5/36
Sum 9: 4/36 = 1/9
Sum 10: 3/36 = 1/12
Sum 11: 2/36 = 1/18
Sum 12: 1/36
(ii) No, the argument is not correct.
Although sums are from 2 to 12 (11 sums), they are not equally likely.
Example: sum 7 has 6 favourable outcomes, while sum 2 has only 1 favourable outcome.
Hence probabilities are different, not 1/11 each.
Q13.
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.
Find the probability that (i) she will buy it (ii) she will not buy it.
Ans:
Total pens = 144
Defective pens = 20
Good pens = 144 − 20 = 124(i) P(buy) = P(good) = 124/144 = 31/36
= 31/36(ii) P(not buy) = P(defective) = 20/144 = 5/36
= 5/36
Q14.
Complete the statements:
(i) P(E) + P(not E) = ____
(ii) Probability of an event that cannot happen is ____ (called ____ event).
(iii) Probability of an event that is certain is ____ (called ____ event).
(iv) Sum of probabilities of all elementary events is ____
(v) Probability of an event is ≥ ____ and ≤ ____
Ans:
(i) 1
(ii) 0, impossible
(iii) 1, sure (certain)
(iv) 1
(v) 0, 1
Q15.
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?
Ans:
Because a fair coin has two outcomes (Head/Tail) and both are equally likely (each has probability 1/2),
so it gives both teams an equal chance.
Q16.
Which of the following cannot be the probability of an event?
(A) 2/3 (B) −1.5 (C) 15% (D) 0.7
Ans:
Probability is always between 0 and 1 (inclusive).
Here, −1.5 is less than 0, so it cannot be a probability.Correct option: (B) −1.5
Q17.
If P(E) = 0.05, what is the probability of ‘not E’?
Ans:
P(not E) = 1 − P(E)
= 1 − 0.05
= 0.95
Q18.
A bag contains lemon flavoured candies only. Malini takes out one candy without looking.
Find probability that she takes out (i) orange flavoured candy (ii) lemon flavoured candy.
Ans:
Since the bag has only lemon candies:
(i) P(orange) = 0 (impossible)
(ii) P(lemon) = 1 (certain)
Q19.
A bag contains 3 red balls and 5 black balls. A ball is drawn at random.
Find probability that the ball drawn is (i) red (ii) not red.
Ans:
Total balls = 3 + 5 = 8
(i) P(red) = 3/8 = 3/8
(ii) P(not red) = 1 − 3/8 = 5/8 = 5/8
Q20.
A box contains 5 red marbles, 8 white marbles and 4 green marbles. One marble is taken out at random.
Find probability that it is (i) red (ii) white (iii) not green.
Ans:
Total marbles = 5 + 8 + 4 = 17
(i) P(red) = 5/17 = 5/17
(ii) P(white) = 8/17 = 8/17
(iii) Not green = red or white = 5 + 8 = 13
P(not green) = 13/17 = 13/17
FAQs – Class 10 Maths Probability
1) Probability kya hoti hai?
Probability kisi event ke hone ki sambhavana ko show karti hai.
Kisi event E ki probability:
P(E) = Number of favourable outcomes / Total number of outcomes
2) Equally likely outcomes ka kya matlab hai?
Jab kisi experiment ke sab outcomes ke chances same hote hain, unhe equally likely outcomes kehte hain.
Example: ek fair coin toss karna (Head ya Tail dono ke chances equal hote hain).
3) Kya probability hamesha 0 aur 1 ke beech hoti hai?
Haan. Kisi bhi event ki probability hamesha 0 se badi ya barabar aur
1 se chhoti ya barabar hoti hai.
0 ka matlab event impossible hai, aur 1 ka matlab event certain hai.
4) ‘Not E’ ki probability kaise nikalte hain?
Agar kisi event E ki probability P(E) di ho, toh
P(not E) = 1 − P(E)
5) Probability chapter boards ke liye kyun important hai?
Probability se direct formula-based aur application-based questions aate hain.
Ye chapter scoring hota hai agar sample space aur favourable outcomes clearly identify kiye jaayein.