Class 10th Mathematics Tamilnadu Board Solution
Exercise 12.1- A ticket is drawn from a bag containing 100 tickets. The tickets are numbered…
- A die is thrown twice. Find the probability of getting a total of 9.…
- Two dice are thrown together. Find the probability that the two digit number…
- Three rotten eggs are mixed with 12 good ones. One egg is chosen at random.…
- Two coins are tossed together. What is the probability of getting at most one…
- One card is drawn randomly from a well shuffled deck of 52 playing cards. Find…
- Three coins are tossed simultaneously. Find the probability of getting (i) at…
- A bag contains 6 white balls numbered from 1 to 6 and 4 red balls numbered from…
- A number is selected at random from integers 1 to 100. Find the probability…
- For a sightseeing trip, a tourist selects a country randomly from Argentina,…
- A box contains 4 Green, 5 Blue and 3 Red balls. A ball is drawn at random.…
- 20 cards are numbered from 1 to 20. One card is drawn at random. What is the…
- A two digit number is formed with the digits 3, 5 and 7. Find the probability…
- Three dice are thrown simultaneously. Find the probability of getting the same…
- Two dice are rolled and the product of the outcomes (numbers) are found. What…
- A jar contains 54 marbles each of which is in one of the colours blue, green…
- A bag consists of 100 shirts of which 88 are good, 8 have minor defects and 4…
- A bag contains 12 balls out of which x balls are white. (i) If one ball is…
- Piggy bank contains 100 fifty-paise coins, 50 one-rupee coins, 20 two-rupees…
Exercise 12.2- If A and B are mutually exclusive events such that P (A) = 3/5 and P (B) = 1/5…
- If A and B are two events such that P (A) = 1/4 , P (B) = 2/5 and P(A ∪ B) =…
- If P (A) = 1/2 , P (B) = 7/10 , P(A ∪ B) = 1. Find (i) P(A ∩ B) (ii) P(A’ ∪…
- If a die is rolled twice, find the probability of getting an even number in the…
- One number is chosen randomly from the integers 1 to 50. Find the probability…
- A bag contains 50 bolts and 150 nuts. Half of the bolts and half of the nuts…
- Two dice are rolled simultaneously. Find the probability that the sum of the…
- A basket contains 20 apples and 10 oranges out of which 5 apples and 3 oranges…
- In a class, 40% of the students participated in Mathematics-quiz, 30% in…
- A card is drawn at random from a well-shuffled deck of 52 cards. Find the…
- A box contains 10 white, 6 red and 10 black balls. A ball is drawn at random.…
- A two digit number is formed with the digits 2, 5, 9 (repetition is allowed).…
- Each individual letter of the word “ACCOMMODATION” is written in a piece of…
- The probability that a new car will get aaward for its design is 0.25, the…
- The probability that A, B and C can solve a problem are 4/5 , 2/3 and 3/7…
Exercise 12.3- If ϕ is an impossible event, then P(ϕ) =A. 1 B. 1/4 C. 0 D. 1/2
- If S is the sample space of a random experiment, then P(S) =A. 0 B. 1/8 C. 1/2…
- If p is the probability of an event A, then p satisfiesA. 0 p 1 B. 0 ≤ p ≤ 1 C.…
- Let A and B be any two events and S be the corresponding sample space. Then p…
- The probability that a student will score centum in mathematics is 4/5 . The…
- If A and B are two events such that P (A) = 0.25, P (B) = 0.05 and P(A ∩ B) =…
- There are 6 defective items in a sample of 20 items. One item is drawn at…
- If A and B are mutually exclusive events and S is the sample space such that p…
- The probabilities of three mutually exclusive events A, B and C are given by…
- If P(A) = 0.25, P (B) = 0.50, P(A ∩ B) = 0.14 then P (neither A nor B) =A.…
- A bag contains 5 black balls, 4 white balls and 3 red balls. If a ball is…
- Two dice are thrown simultaneously. The probability of getting a doublet isA.…
- A fair die is thrown once. The probability of getting a prime or composite…
- Probability of getting 3 heads or 3 tails in tossing a coin 3 times isA. 1/8…
- A card is drawn from a pack of 52 cards at random. The probability of getting…
- The probability that a leap year will have 53 Fridays or 53 Saturdays isA. 2/7…
- The probability that a non-leap year will have 53 Sundays and 53 Mondays isA.…
- The probability of selecting a queen of hearts when a card is drawn from a…
- Probability of sure event isA. 1 B. 0 C. 100 D. 0.1
- The outcome of a random experiment results in either success or failure. If…
- A ticket is drawn from a bag containing 100 tickets. The tickets are numbered…
- A die is thrown twice. Find the probability of getting a total of 9.…
- Two dice are thrown together. Find the probability that the two digit number…
- Three rotten eggs are mixed with 12 good ones. One egg is chosen at random.…
- Two coins are tossed together. What is the probability of getting at most one…
- One card is drawn randomly from a well shuffled deck of 52 playing cards. Find…
- Three coins are tossed simultaneously. Find the probability of getting (i) at…
- A bag contains 6 white balls numbered from 1 to 6 and 4 red balls numbered from…
- A number is selected at random from integers 1 to 100. Find the probability…
- For a sightseeing trip, a tourist selects a country randomly from Argentina,…
- A box contains 4 Green, 5 Blue and 3 Red balls. A ball is drawn at random.…
- 20 cards are numbered from 1 to 20. One card is drawn at random. What is the…
- A two digit number is formed with the digits 3, 5 and 7. Find the probability…
- Three dice are thrown simultaneously. Find the probability of getting the same…
- Two dice are rolled and the product of the outcomes (numbers) are found. What…
- A jar contains 54 marbles each of which is in one of the colours blue, green…
- A bag consists of 100 shirts of which 88 are good, 8 have minor defects and 4…
- A bag contains 12 balls out of which x balls are white. (i) If one ball is…
- Piggy bank contains 100 fifty-paise coins, 50 one-rupee coins, 20 two-rupees…
- If A and B are mutually exclusive events such that P (A) = 3/5 and P (B) = 1/5…
- If A and B are two events such that P (A) = 1/4 , P (B) = 2/5 and P(A ∪ B) =…
- If P (A) = 1/2 , P (B) = 7/10 , P(A ∪ B) = 1. Find (i) P(A ∩ B) (ii) P(A’ ∪…
- If a die is rolled twice, find the probability of getting an even number in the…
- One number is chosen randomly from the integers 1 to 50. Find the probability…
- A bag contains 50 bolts and 150 nuts. Half of the bolts and half of the nuts…
- Two dice are rolled simultaneously. Find the probability that the sum of the…
- A basket contains 20 apples and 10 oranges out of which 5 apples and 3 oranges…
- In a class, 40% of the students participated in Mathematics-quiz, 30% in…
- A card is drawn at random from a well-shuffled deck of 52 cards. Find the…
- A box contains 10 white, 6 red and 10 black balls. A ball is drawn at random.…
- A two digit number is formed with the digits 2, 5, 9 (repetition is allowed).…
- Each individual letter of the word “ACCOMMODATION” is written in a piece of…
- The probability that a new car will get aaward for its design is 0.25, the…
- The probability that A, B and C can solve a problem are 4/5 , 2/3 and 3/7…
- If ϕ is an impossible event, then P(ϕ) =A. 1 B. 1/4 C. 0 D. 1/2
- If S is the sample space of a random experiment, then P(S) =A. 0 B. 1/8 C. 1/2…
- If p is the probability of an event A, then p satisfiesA. 0 p 1 B. 0 ≤ p ≤ 1 C.…
- Let A and B be any two events and S be the corresponding sample space. Then p…
- The probability that a student will score centum in mathematics is 4/5 . The…
- If A and B are two events such that P (A) = 0.25, P (B) = 0.05 and P(A ∩ B) =…
- There are 6 defective items in a sample of 20 items. One item is drawn at…
- If A and B are mutually exclusive events and S is the sample space such that p…
- The probabilities of three mutually exclusive events A, B and C are given by…
- If P(A) = 0.25, P (B) = 0.50, P(A ∩ B) = 0.14 then P (neither A nor B) =A.…
- A bag contains 5 black balls, 4 white balls and 3 red balls. If a ball is…
- Two dice are thrown simultaneously. The probability of getting a doublet isA.…
- A fair die is thrown once. The probability of getting a prime or composite…
- Probability of getting 3 heads or 3 tails in tossing a coin 3 times isA. 1/8…
- A card is drawn from a pack of 52 cards at random. The probability of getting…
- The probability that a leap year will have 53 Fridays or 53 Saturdays isA. 2/7…
- The probability that a non-leap year will have 53 Sundays and 53 Mondays isA.…
- The probability of selecting a queen of hearts when a card is drawn from a…
- Probability of sure event isA. 1 B. 0 C. 100 D. 0.1
- The outcome of a random experiment results in either success or failure. If…
Exercise 12.1
Question 1.A ticket is drawn from a bag containing 100 tickets. The tickets are numbered from one to hundred. What is the probability of getting a ticket with a number divisible by 10?
Answer:From 1 to 100 the numbers divisible by 10 are: 10,20 30 40 50 60 70 80 90 100.
Thus probability of getting a number divisible by 10 
Question 2.A die is thrown twice. Find the probability of getting a total of 9.
Answer:The outcomes on throwing a dice twice are: (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(61),(6,2),(63),(6,4),(6,6),(6,6)
Thus the total probable outcomes = 4
⇒ probability 
Question 3.Two dice are thrown together. Find the probability that the two digit number formed with the two numbers turning up is divisible by 3.
Answer:The outcomes on throwing a dice twice are: (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(61),(6,2),(63),(6,4),(6,6),(6,6)
Thus the total probable outcomes = 12
⇒ probability 
Question 4.Three rotten eggs are mixed with 12 good ones. One egg is chosen at random. What is the probability of choosing a rotten egg?
Answer:The total number of eggs according to question = 12+3
= 15
No of rotten eggs = 3
⇒ probability 
Question 5.Two coins are tossed together. What is the probability of getting at most one head.
Answer:The outcomes when two coins are tossed are (H,H),(T,T),(H,T)(T,H)
H = head, T = tail
No of probable outcomes of at most one head = 3
⇒ probability 
Question 6.One card is drawn randomly from a well shuffled deck of 52 playing cards. Find the probability that the drawn card is
(i) a Diamond (ii) not a Diamond (iii) not an Ace.
Answer:In a deck of 52 cards,13 = diamonds
13 = spade
13 = hearts
13 = club
And there are 4 aces.
(i) ⇒ probability 
(ii) ⇒ probability 
(iii) ⇒ probability 
Question 7.Three coins are tossed simultaneously. Find the probability of getting
(i) at least one head (ii) exactly two tails (iii) at least two heads.
Answer:The outcomes when three coins are tossed aare:(H H H) (H H T) (H T H) (T H H) (T T H) (T H T) (H T T) (T T T)
(i) No of probable outcomes of getting at least one head = 7
⇒ probability 
(ii) no of probable outcomes of getting exactly 2 tails = 3
⇒ probability 
(iii) no of probable outcomes pf at least two heads = 4
⇒ probability 
Question 8.A bag contains 6 white balls numbered from 1 to 6 and 4 red balls numbered from 7 to 10. A ball is drawn at random. Find the probability of getting
(i) an even-numbered ball (ii) a white ball.
Answer:Total no of white balls = 6
Total no of red balls = 4
Thus total no of balls = 6+4 = 10
(i) no of probable outcomes for even numbered ball = 5
⇒ probability 
(ii) no of probable outcomes of a white ball = 6
⇒ probability 
Question 9.A number is selected at random from integers 1 to 100. Find the probability that it is
(i) a perfect square (ii) not a perfect cube.
Answer:(i) no of perfect squares between 1 to 100 = 10
⇒ probability 
(iii) no of perfect cubes from 1 to 100 = 4
⇒ probability 
Question 10.For a sightseeing trip, a tourist selects a country randomly from Argentina, Bangladesh, China, Angola, Russia and Algeria. What is the probability that the name of the selected country will begin with A?
Answer:The probable outcomes for countries name beginning with A = 3
Total no of countries = 6
⇒ probability 
Question 11.A box contains 4 Green, 5 Blue and 3 Red balls. A ball is drawn at random. Find the probability that the selected ball is (i) Red in colour (ii) not Green in colour.
Answer:No of green balls = 4
No of blue balls = 5
No of red balls = 3
Total no of balls = 4+5+3 = 12
(i) no of probable outcomes that the ball is red in colour = 3
⇒ probability 
(ii) no of green balls = 4
⇒ probability 
Question 12.20 cards are numbered from 1 to 20. One card is drawn at random. What is the probability that the number on the card is
(i) a multiple of 4 (ii) not a multiple of 6.
Answer:From 1 to 20, number of numbers divisible by 4 = 5
Total no of outcomes = 20
(i) probable outcomes of numbers divisible by 4 = 5
⇒ probability 
(ii) no of numbers divisible by 6 = 3
⇒ probability 
Question 13.A two digit number is formed with the digits 3, 5 and 7. Find the probability that the number so formed is greater than 57 (repetition of digits is not allowed).
Answer:Since repletion of digits is not allowed, total no of numbers formed with 3,5 7 = 3×2 = 6
No of numbers greater than 57 = 1×2 = 2
⇒ probability 
Question 14.Three dice are thrown simultaneously. Find the probability of getting the same number on all the three dice.
Answer:Total no of outcomes by throwing three dice = 6×6×6 = 256
Probable outcomes of all three same numbers on die = 6
⇒ probability 
Question 15.Two dice are rolled and the product of the outcomes (numbers) are found. What is the probability that the product so found is a prime number?
Answer:The outcomes on throwing a dice twice are: (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(61),(6,2),(63),(6,4),(6,6),(6,6)
Total outcomes = 36
Probable outcome that the product is a prime number = 6
⇒ probability 
Question 16.A jar contains 54 marbles each of which is in one of the colours blue, green and white. The probability of drawing a blue marble is 
and the probability of drawing a green marble is 
. How many white marbles does the jar contain?
Answer:probability of drawing a blue marble is
the probability of drawing a green marble is .
Total no of marbles = 54
Thus no of blue marbles 
No of green marbles
Thus no of white marbles = 54-18-24 = 12
Question 17.A bag consists of 100 shirts of which 88 are good, 8 have minor defects and 4 have major defects. A trader A will accept only the shirt which are good, but the trader B will not accept the shirts which have major defects. One shirt is drawn at random. What is the probability that it is acceptable by (i) A (ii) B?
Answer:Total no of shirts = 100
No of good shirts = 88
No of minor defects = 8
No of major defects = 4
(i) In case of A the probable outcome = 88
⇒ probability 
(ii) in case of B no of probable outcomes = 88+8 = 96
⇒ probability 
Question 18.A bag contains 12 balls out of which x balls are white. (i) If one ball is drawn at random, what is the probability that it will be a white ball. (ii) If 6 more white balls are put in the bag and if the probability of drawing a white ball will be twice that of in (i), then find x.
Answer:Total no of balls = 12
No of white balls = x
(i) If one ball is drawn at random, the probability of it being white 
If 6 more white balls are put in the bag and if the probability of drawing a white ball will be twice that of in(i)
⇒ 
⇒ 
⇒ 2x = 6
⇒ x = 3
Probality 
(i) 
(ii) 3
Question 19.Piggy bank contains 100 fifty-paise coins, 50 one-rupee coins, 20 two-rupees coins and 10 five- rupees coins. One coin is drawn at random. Find the probability that the drawn coin
(i) will be a fifty-paise coin (ii) will not be a five-rupees coin.
Answer:Total no of coins = 100+50+20+10 = 180
No of 50 paise coins = 100
No of one rupee coins = 50
No of two rupees coins = 20
No of five rupees coins = 10
(i) Probable outcome that the coin is 50 paise coin = 100
⇒ probability 
(ii) Probable outcomes that it is a five rupee coin = 10
⇒ probability 
A ticket is drawn from a bag containing 100 tickets. The tickets are numbered from one to hundred. What is the probability of getting a ticket with a number divisible by 10?
Answer:
From 1 to 100 the numbers divisible by 10 are: 10,20 30 40 50 60 70 80 90 100.
Thus probability of getting a number divisible by 10
Question 2.
A die is thrown twice. Find the probability of getting a total of 9.
Answer:
The outcomes on throwing a dice twice are: (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(61),(6,2),(63),(6,4),(6,6),(6,6)
Thus the total probable outcomes = 4
⇒ probability
Question 3.
Two dice are thrown together. Find the probability that the two digit number formed with the two numbers turning up is divisible by 3.
Answer:
The outcomes on throwing a dice twice are: (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(61),(6,2),(63),(6,4),(6,6),(6,6)
Thus the total probable outcomes = 12
⇒ probability
Question 4.
Three rotten eggs are mixed with 12 good ones. One egg is chosen at random. What is the probability of choosing a rotten egg?
Answer:
The total number of eggs according to question = 12+3
= 15
No of rotten eggs = 3
⇒ probability
Question 5.
Two coins are tossed together. What is the probability of getting at most one head.
Answer:
The outcomes when two coins are tossed are (H,H),(T,T),(H,T)(T,H)
H = head, T = tail
No of probable outcomes of at most one head = 3
⇒ probability
Question 6.
One card is drawn randomly from a well shuffled deck of 52 playing cards. Find the probability that the drawn card is
(i) a Diamond (ii) not a Diamond (iii) not an Ace.
Answer:
In a deck of 52 cards,13 = diamonds
13 = spade
13 = hearts
13 = club
And there are 4 aces.
(i) ⇒ probability
(ii) ⇒ probability
(iii) ⇒ probability
Question 7.
Three coins are tossed simultaneously. Find the probability of getting
(i) at least one head (ii) exactly two tails (iii) at least two heads.
Answer:
The outcomes when three coins are tossed aare:(H H H) (H H T) (H T H) (T H H) (T T H) (T H T) (H T T) (T T T)
(i) No of probable outcomes of getting at least one head = 7
⇒ probability
(ii) no of probable outcomes of getting exactly 2 tails = 3
⇒ probability
(iii) no of probable outcomes pf at least two heads = 4
⇒ probability
Question 8.
A bag contains 6 white balls numbered from 1 to 6 and 4 red balls numbered from 7 to 10. A ball is drawn at random. Find the probability of getting
(i) an even-numbered ball (ii) a white ball.
Answer:
Total no of white balls = 6
Total no of red balls = 4
Thus total no of balls = 6+4 = 10
(i) no of probable outcomes for even numbered ball = 5
⇒ probability
(ii) no of probable outcomes of a white ball = 6
⇒ probability
Question 9.
A number is selected at random from integers 1 to 100. Find the probability that it is
(i) a perfect square (ii) not a perfect cube.
Answer:
(i) no of perfect squares between 1 to 100 = 10
⇒ probability
(iii) no of perfect cubes from 1 to 100 = 4
⇒ probability
Question 10.
For a sightseeing trip, a tourist selects a country randomly from Argentina, Bangladesh, China, Angola, Russia and Algeria. What is the probability that the name of the selected country will begin with A?
Answer:
The probable outcomes for countries name beginning with A = 3
Total no of countries = 6
⇒ probability
Question 11.
A box contains 4 Green, 5 Blue and 3 Red balls. A ball is drawn at random. Find the probability that the selected ball is (i) Red in colour (ii) not Green in colour.
Answer:
No of green balls = 4
No of blue balls = 5
No of red balls = 3
Total no of balls = 4+5+3 = 12
(i) no of probable outcomes that the ball is red in colour = 3
⇒ probability
(ii) no of green balls = 4
⇒ probability
Question 12.
20 cards are numbered from 1 to 20. One card is drawn at random. What is the probability that the number on the card is
(i) a multiple of 4 (ii) not a multiple of 6.
Answer:
From 1 to 20, number of numbers divisible by 4 = 5
Total no of outcomes = 20
(i) probable outcomes of numbers divisible by 4 = 5
⇒ probability
(ii) no of numbers divisible by 6 = 3
⇒ probability
Question 13.
A two digit number is formed with the digits 3, 5 and 7. Find the probability that the number so formed is greater than 57 (repetition of digits is not allowed).
Answer:
Since repletion of digits is not allowed, total no of numbers formed with 3,5 7 = 3×2 = 6
No of numbers greater than 57 = 1×2 = 2
⇒ probability
Question 14.
Three dice are thrown simultaneously. Find the probability of getting the same number on all the three dice.
Answer:
Total no of outcomes by throwing three dice = 6×6×6 = 256
Probable outcomes of all three same numbers on die = 6
⇒ probability
Question 15.
Two dice are rolled and the product of the outcomes (numbers) are found. What is the probability that the product so found is a prime number?
Answer:
The outcomes on throwing a dice twice are: (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(61),(6,2),(63),(6,4),(6,6),(6,6)
Total outcomes = 36
Probable outcome that the product is a prime number = 6
⇒ probability
Question 16.
A jar contains 54 marbles each of which is in one of the colours blue, green and white. The probability of drawing a blue marble is and the probability of drawing a green marble is . How many white marbles does the jar contain?
Answer:
probability of drawing a blue marble is
the probability of drawing a green marble is .
Total no of marbles = 54
Thus no of blue marbles
No of green marbles
Thus no of white marbles = 54-18-24 = 12
Question 17.
A bag consists of 100 shirts of which 88 are good, 8 have minor defects and 4 have major defects. A trader A will accept only the shirt which are good, but the trader B will not accept the shirts which have major defects. One shirt is drawn at random. What is the probability that it is acceptable by (i) A (ii) B?
Answer:
Total no of shirts = 100
No of good shirts = 88
No of minor defects = 8
No of major defects = 4
(i) In case of A the probable outcome = 88
⇒ probability
(ii) in case of B no of probable outcomes = 88+8 = 96
⇒ probability
Question 18.
A bag contains 12 balls out of which x balls are white. (i) If one ball is drawn at random, what is the probability that it will be a white ball. (ii) If 6 more white balls are put in the bag and if the probability of drawing a white ball will be twice that of in (i), then find x.
Answer:
Total no of balls = 12
No of white balls = x
(i) If one ball is drawn at random, the probability of it being white
If 6 more white balls are put in the bag and if the probability of drawing a white ball will be twice that of in(i)
⇒
⇒
⇒ 2x = 6
⇒ x = 3
Probality
(i)
(ii) 3
Question 19.
Piggy bank contains 100 fifty-paise coins, 50 one-rupee coins, 20 two-rupees coins and 10 five- rupees coins. One coin is drawn at random. Find the probability that the drawn coin
(i) will be a fifty-paise coin (ii) will not be a five-rupees coin.
Answer:
Total no of coins = 100+50+20+10 = 180
No of 50 paise coins = 100
No of one rupee coins = 50
No of two rupees coins = 20
No of five rupees coins = 10
(i) Probable outcome that the coin is 50 paise coin = 100
⇒ probability
(ii) Probable outcomes that it is a five rupee coin = 10
⇒ probability
Exercise 12.2
Question 1.If A and B are mutually exclusive events such that P (A) = 
and P (B) = 
, then find P(A ∪ B).
Answer:
⇒ 
since mutually exclusive
⇒ 
⇒ 
Question 2.If A and B are two events such that P (A) = 
, P (B) = 
and P(A ∪ B) = 
, then find P(A ∩ B).
Answer:
⇒ 
⇒ 
⇒ 
Required solution,
Question 3.If P (A) = , P (B) = 
, P(A ∪ B) = 1. Find (i) P(A ∩ B) (ii) P(A’ ∪ B’).
Answer:
(i)⇒ 
⇒ 
(ii)
= 
Question 4.If a die is rolled twice, find the probability of getting an even number in the first time or a total of 8.
Answer:The outcomes on throwing a dice twice are: (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(61),(6,2),(63),(6,4),(6,6),(6,6)
Total outcomes = 36
Probable outcome that the even number in the first time or a total of 8.
comes = 20
⇒ probability 
Question 5.One number is chosen randomly from the integers 1 to 50. Find the probability that it is divisible by 4 or 6.
Answer:
Question 6.A bag contains 50 bolts and 150 nuts. Half of the bolts and half of the nuts are rusted. If an item is chosen at random, find the probability that it is rusted or that it is a bolt.
Answer:No of bolts = 50
No of nuts = 150
Total no of things = 150+50 = 200
No of rusted bolts = 25
No of rusted nuts = 75
Probable outcomes = (50)+75
= 125
⇒ probability 
Question 7.Two dice are rolled simultaneously. Find the probability that the sum of the numbers on the faces is neither divisible by 3 nor by 4.
Answer:The outcomes on throwing a dice twice are: (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(61),(6,2),(63),(6,4),(6,6),(6,6)
Total outcomes = 36
Probable outcome that the sum of the numbers on the faces is neither divisible by 3 nor by 4.
comes = 16
⇒ probability 
Question 8.A basket contains 20 apples and 10 oranges out of which 5 apples and 3 oranges are rotten. If a person takes out one fruit at random, find the probability that the fruit is either an apple or a good fruit.
Answer:Total no of fruits = 30
No of apples = 20
No of rotten apples = 5
Thus no of good apples = 15
No of oranges = 10
No of rotten oranges = 3
Thus no of good oranges = 7
probability that the fruit is either an apple or a good fruit 
Question 9.In a class, 40% of the students participated in Mathematics-quiz, 30% in Science-quiz and 10% in both the quiz programmes. If a student is selected at random from the class, find the probability that the student participated in Mathematics or Science or both quiz programmes.
Answer:Let the probability of students participating in mathematics be P(A) ad that in science be P(B).
P(A) = 0.4
P(B) = 0.3
P (A ∩B) = 0.1
P(AUB) = P(A)+P(B)- P(A ∩B)
= 0.4+0.3-0.1
= 0.6
Thus the probability is 
Question 10.A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability that it will be a spade or a king.
Answer:No of spades = 13 out of which 1 I king
Thus no of spades without king = 12
No of kings = 4
Total no of cards = 52
probability that it will be a spade or a king 
Question 11.A box contains 10 white, 6 red and 10 black balls. A ball is drawn at random. Find the probability that the ball drawn is white or red.
Answer:Total no of balls = 26
No of white balls = 10
No of red balls = 6
No of black balls = 10
probability that the ball drawn is white or red 
Question 12.A two digit number is formed with the digits 2, 5, 9 (repetition is allowed). Find the probability that the number is divisible by 2 or 5.
Answer:Since repetition is allowed, the number of 2 digit numbers formed
If a number is divisible by 2 or 5 the unit place should be 2 or 5. Thus probable no of words
Probability 
Question 13.Each individual letter of the word “ACCOMMODATION” is written in a piece of paper, and all 13 pieces of papers are placed in a jar. If one piece of paper is selected at random from the jar, find the probability that
(i) the letter ‘A’ or ‘O’ is selected.
(ii) the letter ‘M’ or ‘C’ is selected.
Answer:Total no of letters = 13
No of A = 2
No of C = 2
No f O = 3
No of M = 2
(i) probability that A or O is selected = 
(ii) probability that M or C is selected = 
Question 14.The probability that a new car will get aaward for its design is 0.25, the probability that it will get an award for efficient use of fuel is 0.35 and the probability that it will get both the awards is 0.15. Find the probability that
(i) it will get atleast one of the two awards
(ii) it will get only one of the awards.
Answer:Let the probability of winning the award for design be P(A) and that for efficient use of fuel be P(B)
P(A) = 0.25
P(B) = 0.35
P(A ∩B) = 0.15
(i) probability that it will get at least one of the two awards
P(AUB) = P(A)+P(B)- P(A ∩B)
= 0.25+0.35-0.15
= 0.45
(ii) probability of getting only oner = P(A)+P(B)-2. P(A ∩B)
= 0.25+0.35-2(0.15)
= 0.3
Question 15.The probability that A, B and C can solve a problem are 
and 
respectively. The probability of the problem being solved by A and B is 
, B and C is 
, A and C is 
. The probability of the problem being solved by all the three is 
. Find the probability that the problem can be solved by atleast one of them.
Answer:Given:
P(A)
P(B)
P(C)
P(A ∩B)
P(B ∩C)
P(A ∩C)
P(A ∩B ∩C)
Thus
P(AUBUC) = P(A)+P(B)+P(C)+ P(A ∩B)+ P(C ∩B)+ P(A ∩C)- P(A ∩B∩C)
If A and B are mutually exclusive events such that P (A) = and P (B) = , then find P(A ∪ B).
Answer:
⇒ since mutually exclusive
⇒
⇒
Question 2.
If A and B are two events such that P (A) = , P (B) = and P(A ∪ B) = , then find P(A ∩ B).
Answer:
⇒
⇒
⇒
Required solution,
Question 3.
If P (A) = , P (B) = , P(A ∪ B) = 1. Find (i) P(A ∩ B) (ii) P(A’ ∪ B’).
Answer:
(i)⇒
⇒
(ii) =
Question 4.
If a die is rolled twice, find the probability of getting an even number in the first time or a total of 8.
Answer:
The outcomes on throwing a dice twice are: (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(61),(6,2),(63),(6,4),(6,6),(6,6)
Total outcomes = 36
Probable outcome that the even number in the first time or a total of 8.
comes = 20
⇒ probability
Question 5.
One number is chosen randomly from the integers 1 to 50. Find the probability that it is divisible by 4 or 6.
Answer:
Question 6.
A bag contains 50 bolts and 150 nuts. Half of the bolts and half of the nuts are rusted. If an item is chosen at random, find the probability that it is rusted or that it is a bolt.
Answer:
No of bolts = 50
No of nuts = 150
Total no of things = 150+50 = 200
No of rusted bolts = 25
No of rusted nuts = 75
Probable outcomes = (50)+75
= 125
⇒ probability
Question 7.
Two dice are rolled simultaneously. Find the probability that the sum of the numbers on the faces is neither divisible by 3 nor by 4.
Answer:
The outcomes on throwing a dice twice are: (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(61),(6,2),(63),(6,4),(6,6),(6,6)
Total outcomes = 36
Probable outcome that the sum of the numbers on the faces is neither divisible by 3 nor by 4.
comes = 16
⇒ probability
Question 8.
A basket contains 20 apples and 10 oranges out of which 5 apples and 3 oranges are rotten. If a person takes out one fruit at random, find the probability that the fruit is either an apple or a good fruit.
Answer:
Total no of fruits = 30
No of apples = 20
No of rotten apples = 5
Thus no of good apples = 15
No of oranges = 10
No of rotten oranges = 3
Thus no of good oranges = 7
probability that the fruit is either an apple or a good fruit
Question 9.
In a class, 40% of the students participated in Mathematics-quiz, 30% in Science-quiz and 10% in both the quiz programmes. If a student is selected at random from the class, find the probability that the student participated in Mathematics or Science or both quiz programmes.
Answer:
Let the probability of students participating in mathematics be P(A) ad that in science be P(B).
P(A) = 0.4
P(B) = 0.3
P (A ∩B) = 0.1
P(AUB) = P(A)+P(B)- P(A ∩B)
= 0.4+0.3-0.1
= 0.6
Thus the probability is
Question 10.
A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability that it will be a spade or a king.
Answer:
No of spades = 13 out of which 1 I king
Thus no of spades without king = 12
No of kings = 4
Total no of cards = 52
probability that it will be a spade or a king
Question 11.
A box contains 10 white, 6 red and 10 black balls. A ball is drawn at random. Find the probability that the ball drawn is white or red.
Answer:
Total no of balls = 26
No of white balls = 10
No of red balls = 6
No of black balls = 10
probability that the ball drawn is white or red
Question 12.
A two digit number is formed with the digits 2, 5, 9 (repetition is allowed). Find the probability that the number is divisible by 2 or 5.
Answer:
Since repetition is allowed, the number of 2 digit numbers formed
If a number is divisible by 2 or 5 the unit place should be 2 or 5. Thus probable no of words
Probability
Question 13.
Each individual letter of the word “ACCOMMODATION” is written in a piece of paper, and all 13 pieces of papers are placed in a jar. If one piece of paper is selected at random from the jar, find the probability that
(i) the letter ‘A’ or ‘O’ is selected.
(ii) the letter ‘M’ or ‘C’ is selected.
Answer:
Total no of letters = 13
No of A = 2
No of C = 2
No f O = 3
No of M = 2
(i) probability that A or O is selected =
(ii) probability that M or C is selected =
Question 14.
The probability that a new car will get aaward for its design is 0.25, the probability that it will get an award for efficient use of fuel is 0.35 and the probability that it will get both the awards is 0.15. Find the probability that
(i) it will get atleast one of the two awards
(ii) it will get only one of the awards.
Answer:
Let the probability of winning the award for design be P(A) and that for efficient use of fuel be P(B)
P(A) = 0.25
P(B) = 0.35
P(A ∩B) = 0.15
(i) probability that it will get at least one of the two awards
P(AUB) = P(A)+P(B)- P(A ∩B)
= 0.25+0.35-0.15
= 0.45
(ii) probability of getting only oner = P(A)+P(B)-2. P(A ∩B)
= 0.25+0.35-2(0.15)
= 0.3
Question 15.
The probability that A, B and C can solve a problem are and respectively. The probability of the problem being solved by A and B is , B and C is , A and C is . The probability of the problem being solved by all the three is . Find the probability that the problem can be solved by atleast one of them.
Answer:
Given:
P(A)
P(B)
P(C)
P(A ∩B)
P(B ∩C)
P(A ∩C)
P(A ∩B ∩C)
Thus
P(AUBUC) = P(A)+P(B)+P(C)+ P(A ∩B)+ P(C ∩B)+ P(A ∩C)- P(A ∩B∩C)
Exercise 12.3
Question 1.If ϕ is an impossible event, then P(ϕ) =
A. 1
B.
C. 0
D.
Answer:P(ϕ) = 0 since ϕ is an impossible event.
Question 2.If S is the sample space of a random experiment, then P(S) =
A. 0
B. 
C.
D. 1
Answer:Since S is the sample space thus P(S) = 1
Question 3.If p is the probability of an event A, then p satisfies
A. 0 < p < 1
B. 0 ≤ p ≤ 1
C. 0 ≤ p < 1
D. 0 < p ≤ 1
Answer:Probability always lies between 0 and 1.
Hence, P(A) = p
⇒ 
Question 4.Let A and B be any two events and S be the corresponding sample space.
Then 
=
A. P(B) – P(A ∩ B)
B. P(A ∩ B) – P(B)
C. P (S)
D. P [(A ∪ B)’]
Answer: = 
As implies the portion of only event B. hence event A and the intersection between these two are not counted.
Question 5.The probability that a student will score centum in mathematics is 
. The probability that he will not score centum is
A. 
B.
C. 
D.
Answer:The probability that a student will score centum in mathematics is .
Thus probability that he will not score centum is
Question 6.If A and B are two events such that
P (A) = 0.25, P (B) = 0.05 and P(A ∩ B) = 0.14, then P(A ∪ B) =
A. 0.61
B. 0.16
C. 0.14
D. 0.6
Answer:
= P (A)+ P (B)- 
⇒ = 0.25+0.05-0.14 = 0.16
Question 7.There are 6 defective items in a sample of 20 items. One item is drawn at random. The probability that it is a non-defective item is
A. 
B. 0
C. 
D. 
Answer:No of defective items = 6
Total no of samples = 20
Probability that the item is not defective = 1-6/20
=
Question 8.If A and B are mutually exclusive events and S is the sample space such that
and S = A ∪ B, then P (A) =
A. 
B. 
C. 
D. 
Answer:Since the event are mutually exclusive the intersection is 0.
Thus P(AUB) = P(A)+P(B)
⇒ P(AUB) = P(A)+3.P(A)
⇒ 1 = 4.P(A)
P(A) =
Question 9.The probabilities of three mutually exclusive events A, B and C are given by 
and
. Then P(A ∪ B ∪ C) is
A. 
B. 
C. 
D. 1
Answer:Since the events are mutually exclusive the intersections between the respective events will be 0.
P(AUBUC) = P(A)+P(B)+P(C)
Question 10.If P(A) = 0.25, P (B) = 0.50, P(A ∩ B) = 0.14 then P (neither A nor B) =
A. 0.39
B. 0.25
C. 0.11
D. 0.24
Answer:P(AUB) = 0.25-0.50-0.14 = 0.61
Thus P (neither A nor B) = 1-P(AUB) = 1-0.61 = 0.39
Question 11.A bag contains 5 black balls, 4 white balls and 3 red balls. If a ball is selected at random, the probability that it is not red is
A.
B. 
C. 
D.
Answer:Total no of balls = 12
No of black balls = 5
No of white balls = 4
No of red balls = 3
Probability thst the ball is not red
Question 12.Two dice are thrown simultaneously. The probability of getting a doublet is
A. 
B. 
C. 
D.
Answer:The outcomes on throwing a dice twice are: (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(61),(6,2),(63),(6,4),(6,6),(6,6)
Total outcomes = 36
Probable outcome of a duplet = 6
⇒ probability
Question 13.A fair die is thrown once. The probability of getting a prime or composite number is
A. 1
B. 0
C. 
D.
Answer:Total outcomes = 6
Thee are 5 numbers which satisfy the prime and composite: 2,3,4,5,6
Thus probability
Question 14.Probability of getting 3 heads or 3 tails in tossing a coin 3 times is
A. 
B.
C.
D.
Answer:The outcomes when three coins are tossed aare:(H H H) (H H T) (H T H) (T H H) (T T H) (T H T) (H T T) (T T T)
No of probable outcomes of getting of 3 heada or 3 tails = 2
⇒ probability 
Question 15.A card is drawn from a pack of 52 cards at random. The probability of getting neither an ace nor a king card is
A. 
B. 
C. 
D. 
Answer:No of ace and king cards = 4+4 = 8
Total no of cards = 52
Thus no of cards neither ace nor king = 52-8 = 44
Probability
Question 16.The probability that a leap year will have 53 Fridays or 53 Saturdays is
A. 
B. 
C. 
D. 
Answer:Leap year has 52 weeks and 2 more days. For 53 Fridays those two should be Thursday, Friday or Friday, Saturday. Hence probability of 53 Fridays 
Similarly probability of 53 Saturdays 
Let these two events be A and B respectively.
Now A and B can't occur simultaneously as there can't be any pair consisting of Friday and Saturday.
So P(A or B) = P(A)+P(B)-P(AB) = 
Question 17.The probability that a non-leap year will have 53 Sundays and 53 Mondays is
A.
B.
C.
D. 0
Answer:Since it is a non leap year thus there cannot be 53 number of Sundays and Mondays each. Thus its probability is 0.
Question 18.The probability of selecting a queen of hearts when a card is drawn from a pack of 52 playing cards is
A. 
B. 
C. 
D. 
Answer:There are 13 hearts of which 1 is queen of hearts out of 52.
probability of selecting a queen of hearts when a card is drawn from a pack of 52 playing cards is
Question 19.Probability of sure event is
A. 1
B. 0
C. 100
D. 0.1
Answer:Since the event is sure, the probability of a sure event = 1.
Question 20.The outcome of a random experiment results in either success or failure. If the probability of success is twice the probability of failure, then the probability of success is
A.
B.
C. 1
D. 0
Answer:Let the probability of failure be x, then the probability of success is 2x.
Thus probability of success = 
If ϕ is an impossible event, then P(ϕ) =
A. 1
B.
C. 0
D.
Answer:
P(ϕ) = 0 since ϕ is an impossible event.
Question 2.
If S is the sample space of a random experiment, then P(S) =
A. 0
B.
C.
D. 1
Answer:
Since S is the sample space thus P(S) = 1
Question 3.
If p is the probability of an event A, then p satisfies
A. 0 < p < 1
B. 0 ≤ p ≤ 1
C. 0 ≤ p < 1
D. 0 < p ≤ 1
Answer:
Probability always lies between 0 and 1.
Hence, P(A) = p
⇒
Question 4.
Let A and B be any two events and S be the corresponding sample space.
Then =
A. P(B) – P(A ∩ B)
B. P(A ∩ B) – P(B)
C. P (S)
D. P [(A ∪ B)’]
Answer:
=
As implies the portion of only event B. hence event A and the intersection between these two are not counted.
Question 5.
The probability that a student will score centum in mathematics is . The probability that he will not score centum is
A.
B.
C.
D.
Answer:
The probability that a student will score centum in mathematics is .
Thus probability that he will not score centum is
Question 6.
If A and B are two events such that
P (A) = 0.25, P (B) = 0.05 and P(A ∩ B) = 0.14, then P(A ∪ B) =
A. 0.61
B. 0.16
C. 0.14
D. 0.6
Answer:
= P (A)+ P (B)-
⇒ = 0.25+0.05-0.14 = 0.16
Question 7.
There are 6 defective items in a sample of 20 items. One item is drawn at random. The probability that it is a non-defective item is
A.
B. 0
C.
D.
Answer:
No of defective items = 6
Total no of samples = 20
Probability that the item is not defective = 1-6/20
=
Question 8.
If A and B are mutually exclusive events and S is the sample space such that and S = A ∪ B, then P (A) =
A.
B.
C.
D.
Answer:
Since the event are mutually exclusive the intersection is 0.
Thus P(AUB) = P(A)+P(B)
⇒ P(AUB) = P(A)+3.P(A)
⇒ 1 = 4.P(A)
P(A) =
Question 9.
The probabilities of three mutually exclusive events A, B and C are given by and. Then P(A ∪ B ∪ C) is
A.
B.
C.
D. 1
Answer:
Since the events are mutually exclusive the intersections between the respective events will be 0.
P(AUBUC) = P(A)+P(B)+P(C)
Question 10.
If P(A) = 0.25, P (B) = 0.50, P(A ∩ B) = 0.14 then P (neither A nor B) =
A. 0.39
B. 0.25
C. 0.11
D. 0.24
Answer:
P(AUB) = 0.25-0.50-0.14 = 0.61
Thus P (neither A nor B) = 1-P(AUB) = 1-0.61 = 0.39
Question 11.
A bag contains 5 black balls, 4 white balls and 3 red balls. If a ball is selected at random, the probability that it is not red is
A.
B.
C.
D.
Answer:
Total no of balls = 12
No of black balls = 5
No of white balls = 4
No of red balls = 3
Probability thst the ball is not red
Question 12.
Two dice are thrown simultaneously. The probability of getting a doublet is
A.
B.
C.
D.
Answer:
The outcomes on throwing a dice twice are: (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)
(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)
(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)
(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)
(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)
(61),(6,2),(63),(6,4),(6,6),(6,6)
Total outcomes = 36
Probable outcome of a duplet = 6
⇒ probability
Question 13.
A fair die is thrown once. The probability of getting a prime or composite number is
A. 1
B. 0
C.
D.
Answer:
Total outcomes = 6
Thee are 5 numbers which satisfy the prime and composite: 2,3,4,5,6
Thus probability
Question 14.
Probability of getting 3 heads or 3 tails in tossing a coin 3 times is
A.
B.
C.
D.
Answer:
The outcomes when three coins are tossed aare:(H H H) (H H T) (H T H) (T H H) (T T H) (T H T) (H T T) (T T T)
No of probable outcomes of getting of 3 heada or 3 tails = 2
⇒ probability
Question 15.
A card is drawn from a pack of 52 cards at random. The probability of getting neither an ace nor a king card is
A.
B.
C.
D.
Answer:
No of ace and king cards = 4+4 = 8
Total no of cards = 52
Thus no of cards neither ace nor king = 52-8 = 44
Probability
Question 16.
The probability that a leap year will have 53 Fridays or 53 Saturdays is
A.
B.
C.
D.
Answer:
Leap year has 52 weeks and 2 more days. For 53 Fridays those two should be Thursday, Friday or Friday, Saturday. Hence probability of 53 Fridays
Similarly probability of 53 Saturdays
Let these two events be A and B respectively.
Now A and B can't occur simultaneously as there can't be any pair consisting of Friday and Saturday.
So P(A or B) = P(A)+P(B)-P(AB) =
Question 17.
The probability that a non-leap year will have 53 Sundays and 53 Mondays is
A.
B.
C.
D. 0
Answer:
Since it is a non leap year thus there cannot be 53 number of Sundays and Mondays each. Thus its probability is 0.
Question 18.
The probability of selecting a queen of hearts when a card is drawn from a pack of 52 playing cards is
A.
B.
C.
D.
Answer:
There are 13 hearts of which 1 is queen of hearts out of 52.
probability of selecting a queen of hearts when a card is drawn from a pack of 52 playing cards is
Question 19.
Probability of sure event is
A. 1
B. 0
C. 100
D. 0.1
Answer:
Since the event is sure, the probability of a sure event = 1.
Question 20.
The outcome of a random experiment results in either success or failure. If the probability of success is twice the probability of failure, then the probability of success is
A.
B.
C. 1
D. 0
Answer:
Let the probability of failure be x, then the probability of success is 2x.
Thus probability of success =