CBSE Class 10 Maths HOTs Probability

PROBABILITY

1.  An integer is chosen at random from the first two hundreds digit. What is the

probability that the integer chosen is divisible by 6 or 8.                  (Ans : 1/4 )

Ans: Multiples of 6 first 200 integers

6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114,

120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192, 198

Multiples of 8 first 200 integers

8,16,24,32,40,48,56,64,72,80,88,96,104,112,120,128,136,144,152,160,

168, 176,184,192,200

Number of Multiples of 6 or 8 = 50

P(Multiples of 6 or 8) = 50 / 200 = 1/4

2.  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: x = 3)

Ans: Random drawing of balls ensures equally likely outcomes

Total number of balls = 12

Total number pf possible outcomes = 12

Number of black balls = x

(1) out of total 12 outcomes, favourable outcomes = x

b (black ball) = Number of favourable outcomes      = x/12

Total number of possible outcomes

(2) if 6 more black balls are put in the bag, then

The total number of black balls = x + 6

Total number of balls in the bag = 12 + 6 = 18

According to the question

Probability of drawing black ball is second case

= 2 X probability drawing of black ball in first case

hence number of black balls = 3

Question. A bag contains 8 red balls and x blue balls, the odd against drawing a blue ball are 2: 5. What is the value of x? 
Ans: No. of blue balls be x
No. of red balls be 8
Total no. of balls = x + 8
Probability of drawing blue balls = x/8+x 

Question. A card is drawn from a well shuffled deck of cards
(i) What are the odds in favour of getting spade? (Ans: 1:3, 3:1, 3:10, 1:25)
(ii) What are the odds against getting a spade?
(iii) What are the odds in favour of getting a face card?
(iv) What are the odds in favour of getting a red king
Ans: Total cards 52
Spade = 13
Remaining cards 39

i) The odds in favour of getting spade 13
The odds is not in favour of getting spade 39
= 13/52 : 39/52 = 1 : 3

ii) The odds against getting a spade 39
The odds not against getting a spade 13 
= 13/52 : 39/52 = 3 : 1

iii) The odds in favour of getting a face card 12
The odds not in favour of getting a face card 40
= 12/52 : 40/52 = 3 :10
iv) The odds in favour of getting a red king 2
The odds not in favour of getting a red king 50
= 2/52 : 50/52 = 1:25

Question. A die is thrown repeatedly until a six comes up. What is the sample space for this experiment? HINT ;A= {6} B={1,2,3,4,5,}
Ans: The sample space is = {A, BA, BBA, BBBA, BBBBA……..})

Question. Why is tossing a coin considered to be a fair way of deciding which team should get the ball at the beginning of a foot ball match?
Ans: equally likely because they are mutually exclusive events .

Question. 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 the number of blue balls is the bag be x
Then total number of balls is the bag = 5 + x
∴ Number of all possible outcomes = 5 + x
Number of outcomes favourable to the event of drawing a blue ball = x
(Q there are x blue balls)
∴ Probability of drawing a blue ball x/5+x
Similarly, probability of drawing a red ball = 5/5+x
According to the answer x/5+x = 2 (5/5+x)
x = 10

Question. 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: Number of all possible outcomes = 12
Number of outcomes favourable to the event of drawing black ball = x
Required probability = x/12
Now when 6 more black balls are put in the box,
Number of all possible outcomes = 12 + 6 = 18
Number of outcomes favourable to the event of drawing a black ball = x + 6
∴ Probability of drawing a black ball = x+6/18
According to the question,
x+6 = 2 (x/12)
∴ x = 3

Question. If 65% of the populations have black eyes, 25% have brown eyes and the remaining have blue eyes. What is the probability that a person selected at random has (i) Blue eyes (ii) Brown or black eyes (iii) Blue or black eyes (iv) neither blue nor brown eyes
Ans: No. of black eyes = 65
No. of Brown eyes = 25
No. of blue eyes = 10
Total no. of eyes = 180
i) P (Blue eyes) = 10/100 = 1/10
ii) P (Brown or black eyes) = 90/100 = 9/10
iii) P(Blue or black eyes) = 75/100 = 3/4
iv) P(neither blue nor brown eyes) = 65/100 = 13/20

Question. Find the probability of having 53 Sundays in
(i) a leap year (ii) a non leap year
Ans: An ordinary year has 365 days i.e. 52 weeks and 1 day
This day can be any one of the 7 days of the week.
∴ P(that this day is Sunday) = 1/7
Hence, P(an ordinary year has 53 Sunday) = 1/7
A leap year 366 days i.e. 52 weeks and 2 days
This day can be any one of the 7 days of the week
∴  P (that this day is Sunday) = 2/7
Hence, P(a leap year has 53 Sunday) = 2/7

Question. Find the probability that the month June may have 5 Mondays in
(i) a leap year (ii) a non leap year 
Ans: 2/7 ,2/7

Self Practice

Question. Find the probability that the month February may have 5 Wednesdays in
(i) a leap year (ii) a non leap year
Ans: 1/7 , 0

Self Practice

Question. Five cards – the ten, jack, queen, king and ace, are well shuffled with their face downwards. One card is then picked up at random.
(i) What is the probability that the card is a queen?
(ii) If the queen is drawn and put aside, what is the probability that the second card picked up is a (a) an ace (b) a queen
Ans : Here, the total number of elementary events = 5
(i) Since, there is only one queen
∴  Favourable number of elementary events = 1
∴  Probability of getting the card of queen = 1/5
(ii) Now, the total number of elementary events = 4
(a) Since, there is only one ace
∴ Favourable number of elementary events = 1 
∴ Probability of getting an ace card = 1/4
(b) Since, there is no queen (as queen is put aside)
∴ Favourable number of elementary events = 0
∴ Probability of getting a queen = 0/4 = 0

Question. A number x is chosen at random from the numbers -3, -2, -1, 0 1, 2, 3. What is the probability that |x| < 2 
Ans : x can take 7 values
To get |x| <2 take –1, 0, 1
Probability (| x | < 2) = 3/7

Question. A number x is selected from the numbers 1,2,3 and then a second number y is randomly selected from the numbers 1,4,9. What is the probability that the product xy of the two numbers will be less than 9?
Ans : Number X can be selected in three ways and corresponding to each such way
there are three ways of selecting number y . Therefore , two numbers can be selected in 9 ways as listed below:
(1,1), (1,4), (2,1), (2,4), (3,1)
∴ Favourable number of elementary events = 5
Hence, required probability = 5/9

Question. In the adjoining figure a dart is thrown at the dart board and lands in the interior of the circle. What is the probability that the dart will land in the shaded region.  

Ans: We have
AB = CD = 8 and AD = BC = 6
using Pythagoras Theorem is DABC, we have
AC² = AB² + BC²
AC² = 82 + 62 = 100
AC = 10
OA = OC = 5 [ Q O is the midpoint of AC]
∴ Area of the circle = p (OA)2 = 25 π sq units [Q Area = πr²]
Area of rectangle ABCD = AB x BC = 8 x 6 = 48 sq units
Area of shaded region = Area of the circle – Area of rectangle ABCD
Area of shaded region = 25 π - 48 sq unit.
Hence
P (Dart lands in the shaded region) = Area of shaded region/Area of circle = 25π - 48/25π

Question. In the fig points A ,B ,C and D are the centres of four circles ,each having a radius of 1 unit . If a point is chosen at random from the interior of a square ABCD ,what is the probability that the point will be chosen from the shaded region . 

Ans: Radius of the circle is 1 unit
Area of the circle = Area of 4 sector
πr2 = π x 12 = p
Side of the square ABCD = 2 units
Area of square = 2 x 2 = 4 units
Area shaded region is
= Area of square – 4 x Area of sectors
= 4 - π
Probability = (4 - Π/4)

Question. In the adjoining figure ABCD is a square with sides of length 6 units points P & Q are the mid points of the sides BC & CD respectively. If a point is selected at random from the interior of the square what is the probability that the point will be chosen from the interior of the triangle APQ. 

Ans: Area of triangle PQC = 1/2 x 3 x 3 = 9/2 = 4. 5 units
Area of triangle ABP = 1/2 x 6 x 3 = 9
Area of triangle ADQ = 1/2 x 6 x 3 = 9
Area of triangle APQ = Area of a square – (Area of a triangle PQC + Area of
triangle
ABP + Area of triangle ABP)
= 36 – (18+4.5)
= 36 – 22.5
= 13.5
Probability that the point will be chosen from the interior of the triangle APQ = 13.5/36
                                                                                                                         = 135/360 = 3/8

Question. In a musical chair game the person playing the music has been advised to stop playing the music at any time within 2 minutes after she starts playing. What is the probability that the music will stop within the half minute after starting.
Ans: Here the possible outcomes are all the numbers between 0 and 2.
This is the portion of the number line from 0 to 2 as shown in figure.
Let A be the event that ‘the music is stopped within the first half minute.’ Then,
outcomes favorable to event A are all points on the number line from O to Q i.e., from 0 to 1/2 . 

The total number of outcomes are the points on the number line from O to P i.e., 0 to 2.
∴ P (A) =Length of OQ/Length of OP = 1 / 2/2 = 1/4

Question. A jar contains 54 marbles each of which is blue , green or white. The probability of selecting a blue marble at random from the jar is 1/3 and the probability of selecting a green marble at random is 4/9 . How many white marbles does the jar contain?
Ans: Let there be b blue, g green and w white marbles in the marbles in the jar. Then,
b + g + w = 54
∴ P (Selecting a blue marble) = b/54
It is given that the probability of selecting a blue marble is 1/3
∴ 1/3 = b/54 => b = 18
We have,
P(Selecting a green marble) = 4/9
=> g/54 = 4/9 [Q P (Selecting a green marble) = 4/9 (Given)]
=> g = 24
Substituting the values of b and g in (i), we get
18 + 24 + w = 54 => w = 12

More MCQs for NCERT Class 10 Mathematics Probability.......

1. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 5?

(A) 1/2

(B) 2/5

(C) 8/15

(D) 9/20

Answer : (D)

2. A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?

(A) 10/21

(B) 11/21

(C) 2/7

(D) 5/7

Answer : (A)

3. In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked up randomly. What is the probability that it is neither red nor green?

(A) 1/3

(B) 3/4

(C) 7/19

(D) 8/21

Answer : (A)

4. What is the probability of getting a sum 9 from two throws of a dice?

(A) 1/6

(B) 1/8

(C) 1/9

(D) 1/12

Answer : (C)

5. Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is even?

(A) 1/2

(B) 3/4

(C) 3/8

(D) 5/16

Answer : (B)

6. In a class, there are 15 boys and 10 girls. Three students are selected at random. The probability that 1 girl and 2 boys are selected, is:

(A) 21/46

(B) 25/117

(C) 1/50

(D) 3/25

Answer : (A)

7. From a pack of 52 cards, two cards are drawn together at random. What is the probability of both the cards being kings?

(A) 1/15

(B) 25/57

(C) 35/256

(D) 1/221

Answer : (D)

8.Two dice are tossed. The probability that the total sum is a prime number is:

(A) 1/6

(B) 5/12

(C) 1/2

(D) 7/9

Answer : (B)

9. A card is drawn from a pack of 52 cards. The probability of getting a queen of club or a king of heart is:

(A) 1/13

(B) 2/13

(C) 1/26

(D) 1/52

Answer : (C)

10. Two cards are drawn together from a pack of 52 cards. The probability that one is a spade and other is a heart, is :

(A) 3/20

(B) 29/34

(C) 47/100

(D) 13/102

Answer : (D)

11. One card is drawn at random from a pack of 52 cards. What is the probability that the card drawn is a face card (Jack, Queen and King only)?

(A) 1/13 

(B) 3/13

(C) 1/4

(D) 9/ 52

Answer : (B)

12. There are 30 cards of the same size in a bag on which natural numbers 1 to 30 are written. One card is taken out of the bag at random. Then the probability that the number on the selected card is not divisible by 3 is

 (A) 1/3

(B) 3/4

(C) 2/3

(D) 1/4

Answer : (C)

13. Three fair dice are rolled, then the probability that the same number will appear on each of them is

(A) 1/6

(B) 1/18

(C) 1/36

(D) 3/28

Answer : (C)

14. Three coins are tossed, the probability of getting at most 2 heads is

(A) 3/8

(B) 1/2

(C) 7/8

(D) 1/8

Answer : (C)

15. A dice is tossed 100 times and the data is recorded as below
Outcomes   1    2    3    4   5   6
Frequency 20  15  20   15  20 10

The probability that we get at even number in a trial is

(A) 2/5

(B) 3/5

(C) 1/5

(D) 4/5

Answer : (A)

16. A letter is chosen at random from the word “probability”. The probability that it is a vowel is

(A) 1/11

(B) 2/11

(C) 3/11

(D) 4/11

Answer : (D)

17. The probability of guessing the correct answer to a certain question is p/12. If the probability of not guessing the correct answer to the same question is 3/4, the value of p is

(A) 3 

(B) 4

(C) 2 

(D) 1

Answer : (A)

18. Sum of probabilities of all the events in a sample space related to any event is

(A) 1 

(B) 0

(C) –1 

(D) not defined

Answer : (A)

19. Probability that a non leap year should have 53 Mondays, will be

(A) 2/7

(B) 3/7

(C) 1/7

(D) 5/7

Answer : (C)

20. A bag contains 10 red balls and some white balls. If the probability of drawing a white ball is double that of a red ball, then number of white balls in the bag will be

(A) 10 

(B) 15 

(C) 20 

(D) 25

Answer : (C)

21. Each outcome of a sample space related to any random experiment is known as

(A) compound event 

(B) elementary event

(C) sure event 

(D) impossible event

Answer : (B)

22. If all the face cards are removed from a pack of 52 cards and then a card is randomly drawn then the probability of getting a ‘10 of heart’ will be

(A) 1/40

(B) 2/49

(C) 3/40

(D) 3/17

Answer : (A)

23 Box A contains 30% first grade articles. Box B contains 40% first grade articles. One article is drawn from each box. Then the probability that both articles drawn are first grade is

(A) 1/25 

(B) 3/25 

(C) 7/25 

(D) 9/25

Answer : (B)

24. A box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, then the probability that it bears a two-digit number is :

(A) 9/10 

(B) 7/10 

(C) 3/5 

(D) 2/5.

Answer : (A) 

25. 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.

(A) 1/4 

(B) 1

(C) 3/4 

(D) 0.

Answer : (C) 

26. Three unbiased coins are tossed together. Then the probability of getting at least one head and one tail is

(A) 1/4 

(B) 1

(C) 3/4 

(D) 0.

Answer : (C)

27. When three coins are tossed together the possible outcomes are HHH, HHT, HTH, THH, HTT, THT, TTH and TTT.

Answer : (A)

28. A die is thrown twice. Then the probability that 5 will come up at least once is

(A) 11/36 

(B) 7/36

(C) 5/36 

(D) 0.

Answer : (B)

29. Ina single throw of two dice,the probability of getting a doublet of odd numbers is
(A) 11/12 

(B) 1/12

(C) 5/12 

(D) 5/6.

Answer :(C) 

30. In a single throw of three dice, the probability of getting a total of 17 or 18 is

(A) 53/54 

(B) 51/54

(C) 1/54 

(D) 0.

Answer : (B)

31. Mallica and Deepica are friends. Then the probability that both have same birthday is (ignoring a leap year)

(A) 364/365 

(B) 1/365

(C) 363/365 

(D) 2/365

Answer : (A)

32. Mallica and Deepica are friends. Then the probability that both have different birthdays is
(A) 364/365 

(B) 1/365

(C) 363/365 

(D) 2/365

Answer : (B)

33. A card is drawn at random from a pack of 52 playing cards. Then the probability that the card is neither an ace nor a king is
(A) 10/13 

(B) 11/13

(C) 7/13 

(D) 9/13.

Answer : (C)

34. From a well shuffled pack of 52 cards, black aces and black queens are removed and from the remaining cards, a card is drawn at random. Then the probability of drawing a king or a queen is

(A) 7/8 

(B) 3/4

(C) 1/8 

(D) 1/2.

Answer : (C)

35. The king, queen and jack of hearts are removed from a deck of 52 playing cards and then well shuffled. One card is selected from the remaining cards. Then the probability of getting a king is

(A) 1/49 

(B) 2/49 

(C) 3/49 

(D) 1

Answer : 

Please refer to link below to download pdf file of CBSE Class 10 Probability HOTs (1)

Please refer to attached file for CBSE Class 10 Mathematics HOTs Probability

HOTS ON PROBABILITY

ADDITIONAL QUESTION

Probability

1. Two dice are thrown once. What is the probability of getting a doublet?

2. A jar contains 54 marbles of colour blue, green and white. The probability of selecting a blue marble at random from the jar is 1/3 and the probability of selecting a green marble at random is 4/9. How many white marbles do the jar contains?

3. In a leap year, find the probability that there are 53 Sundays in the year.

4. A letter is chosen at random from the word MISSISSIPPI. Find the probability of getting i) a vowel. ii) a consonant.

5. A bag contains 4 whit balls, 6 red balls and 7 black balls. One ball is drawn at random from the bag. What is the probability that the ball drawn is i) not a black ball ii) neither white nor black iii) red or white.

6. A box has cards numbered from 20 to 100. Cards are mixed thoroughly and a card is drawn from the box at random. Find the probability that the number on the card drawn from the box is i) an odd number ii) a perfect square number and iii) a number divisible by 7.

Please refer to link below for CBSE Class 10 Mathematics HOTs Probability Set A

PROBABLITY

1) A die is thrown once. Find the probability of getting multiple of 2 or 3.

2) Find the probability of getting 53 Fridays in a leap year.

3) Write a sample space when two coins are tossed simultaneously?

4) A bag contain 4 red, 5 black and 6 white balls & ball is drawn from the bag at random.

Find the probability that ball drawn is

a) Not black b) Red or White

5) In a single throw of two dice, find the probability of not getting the same number on the two dice.

6) The probability that it will rain tomorrow is 0.85. What is the probability that it will not rain tomorrow ?

7) What is the probability of

a) A sure event b) Impossible event

8) A card is drawn at random from a well shuffled pack of playing cards.

Find the probability that the card drawn is

a) Red and a queen b) Either a king or jack

9) Two coins are tossed simultaneously once. What is the probability of getting

a) Exactly one head b) No head

10) 15 cards numbered 1,2,3………15 are put in a box and mixed thoroughly & card is drawn at random from the box. Find the probability that the card drawn is

a) neither prime nor odd b) a number which is a perfect square.