CLASS 10 · MATHEMATICS · CHAPTER 14
Probability
Count equally likely outcomes, use complementary events and explain why each probability lies between 0 and 1.
25 worked answersAll 1 exercise sets
What you will learn
- Calculate theoretical probability from favourable and total outcomes.
- Use P(not E) = 1 − P(E).
- Check whether outcomes are genuinely equally likely.
25 worked answers · short steps first, explanations when you need them
Exercise 14.1
Textbook pages 214–217
Question 1
Complete the five statements about complementary, impossible, certain and elementary events and the range of probability.
(i) P(E) + P(not E) = 1
(ii) 0; impossible event. (iii) 1; sure or certain event.
(iv) The elementary probabilities sum to 1. (v) 0 ≤ P(E) ≤ 1.
Answer
The blanks are 1; 0, impossible; 1, sure/certain; 1; 0 and 1.
Question 2
Decide which experiments have equally likely outcomes: a car starting, a basketball shot, a true-false answer being right, and a baby's sex.
(i) No. (ii) No. (iii) No. (iv) Yes, under the textbook assumption.
Answer
Only experiment (iv) is treated as having equally likely outcomes.
Question 3
Why is a coin toss a fair way to decide which football team gets the ball?
A fair coin has two equally likely outcomes: head and tail.
Answer
Each team has probability 1/2.
Question 4
Which cannot be a probability: 2/3, −1.5, 15% or 0.7?
A probability must satisfy 0 ≤ P ≤ 1.
−1.5 < 0
Question 5
If P(E) = 0.05, find P(not E).
P(not E) = 1 − 0.05 = 0.95
Question 6
A bag contains only lemon candies. Find the probabilities of drawing (i) an orange candy and (ii) a lemon candy.
Orange is impossible; lemon is certain.
Question 7
The probability that two students do not have the same birthday is 0.992. Find the probability that they have the same birthday.
P(same) = 1 − 0.992 = 0.008
Question 8
A bag has 3 red and 5 black balls. Find the probabilities of drawing (i) red and (ii) not red.
Total = 8; P(red) = 3/8; P(not red) = 5/8
Question 9
A box has 5 red, 8 white and 4 green marbles. Find P(red), P(white) and P(not green).
Total = 17
P(red) = 5/17; P(white) = 8/17; P(not green) = 13/17
Answer
5/17, 8/17 and 13/17.
Question 10
A bank has 100 fifty-paise, 50 one-rupee, 20 two-rupee and 10 five-rupee coins. Find P(50p) and P(not ₹5).
Total coins = 180
P(50p) = 100/180 = 5/9
P(not ₹5) = 170/180 = 17/18
Question 11
A tank has 5 male and 8 female fish. Find the probability that a randomly selected fish is male.
P(male) = 5/(5 + 8) = 5/13
Question 12
A fair spinner has numbers 1 to 8. Find the probabilities of (i) 8, (ii) odd, (iii) greater than 2 and (iv) less than 9.
(i) 1/8
(ii) 4/8 = 1/2
(iii) 6/8 = 3/4
(iv) 8/8 = 1
Answer
1/8, 1/2, 3/4 and 1.
Question 13
A fair die is thrown. Find the probabilities of (i) a prime, (ii) a number between 2 and 6 and (iii) an odd number.
Primes: 2,3,5; between 2 and 6: 3,4,5; odd: 1,3,5.
Each probability = 3/6 = 1/2
Answer
- (i) ,
- (ii) and
- (iii) are each 1/2.
Question 14
From a 52-card deck, find P(red king), P(face card), P(red face card), P(jack of hearts), P(spade) and P(queen of diamonds).
Counts: 2, 12, 6, 1, 13, 1 out of 52
Answer
1/26, 3/13, 3/26, 1/52, 1/4 and 1/52.
Question 15
Five diamond cards—10, J, Q, K and A—are shuffled. Find P(queen). After setting aside a drawn queen, find P(ace) and P(queen) on the next draw.
First P(queen) = 1/5
After removing Q: P(ace) = 1/4; P(queen) = 0
Answer
- (i) 1/5;
- (ii) (a) 1/4, (b) 0.
Question 16
Twelve defective pens are mixed with 132 good pens. Find the probability of selecting a good pen.
P(good) = 132/(132 + 12) = 132/144 = 11/12
Question 17
A lot has 20 bulbs, 4 defective. Find P(first defective). If a good bulb is drawn and not replaced, find P(next good).
(i) 4/20 = 1/5
(ii) After one good is removed: 15 good among 19, so 15/19
Question 18
A box has discs numbered 1 to 90. Find P(two-digit), P(perfect square) and P(divisible by 5).
Two-digit: 10 to 90 = 81 numbers, so 81/90 = 9/10
Squares: 1² to 9² = 9, so 9/90 = 1/10
Multiples of 5: 18, so 18/90 = 1/5
Answer
9/10, 1/10 and 1/5.
Question 19
A die has faces A, B, C, D, E, A. Find P(A) and P(D).
A appears on 2 of 6 faces; D on 1 of 6
Answer
P(A) = 1/3; P(D) = 1/6.
Question 20 ★
A die is dropped randomly on a 3 m by 2 m rectangle containing a circle of diameter 1 m. Find the probability that it lands inside the circle.
Rectangle area = 3×2 = 6 m²
Circle area = π(1/2)² = π/4 m²
Probability = (π/4)/6 = π/24
Answer
π/24, approximately 0.131.
Question 21
Of 144 pens, 20 are defective. Nuri buys a randomly offered pen only if it is good. Find P(buy) and P(not buy).
Good = 144 − 20 = 124
P(buy) = 124/144 = 31/36; P(not buy) = 20/144 = 5/36
Question 22
Complete the probabilities for sums 2 through 12 on two dice, then assess the claim that each sum has probability 1/11.
For sums 2–12, favourable counts are 1,2,3,4,5,6,5,4,3,2,1 out of 36.
The sums are not equally likely, so the 1/11 claim is false.
Answer
Probabilities: 1/36, 2/36, 3/36, 4/36, 5/36, 6/36, 5/36, 4/36, 3/36, 2/36, 1/36; claim rejected.
Question 23
A coin is tossed three times. Hanif wins only with HHH or TTT. Find the probability that he loses.
Total outcomes = 2³ = 8; winning outcomes = 2
P(lose) = (8 − 2)/8 = 3/4
Question 24
A die is thrown twice. Find P(5 appears neither time) and P(5 appears at least once).
P(no 5 twice) = 5/6 × 5/6 = 25/36
P(at least one 5) = 1 − 25/36 = 11/36
Question 25
Assess two arguments: three outcome labels for two coins each have probability 1/3; and odd/even on a die each have probability 1/2.
(i) Incorrect: HH and TT each have probability 1/4, while one of each has probability 2/4 = 1/2.
(ii) Correct: three odd and three even faces give probability 3/6 = 1/2.
Answer
Argument
- (i) is incorrect; argument
- (ii) is correct.