15. PROBABILITY (SCERT)

SEBA Class 10 Chapter 15. PROBABILITY

Chapter 15. PROBABILITY

Chapter 15. Probability

Exercise 15.1 Complete solutions

Important Notes :

1. The theoretical (classical) probability of an event E , written as P(E) , is defined as

2. The probability of a sure event or certain event is 1.
3. The probability of an impossible event is 0.
4. The probability of an event is a number such that
5. An event having only one outcome of the experiment is called an elementary event .

6. The sum of the probabilities of all the elementary events of an experiment is 1.
7. For any event , , where stands for ‘not ’. and are called complementary events.

8. For one coin :

The sample space H , T

9. For two coins :

The sample space HH , HT , TH , TT

10. For three coins :

The sample space HHH , HHT , HTT , HTH , THT , THH , TTH , TTT

11. For one dice :

The sample space 1, 2 , 3 , 4 , 5 , 6

12. For two dice :

The sample space (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),(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)

13. The playing card consists of 52 cards which are divided into 4 suits of 13 cards each :
(i) Spades (black colour) : Ace , king , queen , jack , 10 ,9 , 8 , 7 , 6 , 5 , 4 , 3 and 2

(ii) Clubs (black colour) : Ace , king , queen , jack , 10 ,9 , 8 , 7 , 6 , 5 , 4 , 3 and 2

(iii) Hearts (Red colour) : Ace , king , queen , jack , 10 ,9 , 8 , 7 , 6 , 5 , 4 , 3 and 2

(iv) Diamonds (Red colour) : Ace , king , queen , jack , 10 ,9 , 8 , 7 , 6 , 5 , 4 , 3 and 2

(v) Kings, queens and jacks are called face cards.

14. Prime number 1 to 100 are : 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 47 , 53 , 59 , 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97 .

Class 10 Maths Chapter 15. PROBABILITY Exercise 15.1 Solutions

1. Complete the following statements :

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

(ii) The probability of an event that is certain to 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 .

Solution: (i) 1

[ We have , ]

(ii) 0 , impossible event

(iii) 1 , sure event or certain event .

(iv) 1

[ The sum of the probabilities of all the elementary events of an experiment is 1 .]

(v) 0 , 1

[The probability of an event is greater than or equal to 0 and less than or equal to 1 .]

2. Which of the 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 .

Solution : (i) The experiments have no equally likely outcomes . Because, we can not justify to assume that each outcome is a likely to occur as of the other .

(ii) The experiments have no equally likely outcomes . Because, we can not justify to assume that each outcome is a likely to occur as of the other .

(iii) The experiments have equally likely outcomes . Because, we can justify to assume that each outcome is a likely to occur as of the other .

(iv) The experiments have equally likely outcomes .Because, we can justify to assume that each outcome is a likely to occur as of the other .

3. 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 ?

Solution : When we toss a coin, the outcomes head and tail are equally likely . So, the result of an individual coin toss is completely unpredictable .

4. Which of the following cannot be the probability of an event ?

(A) (B) (C) (D)

Solution : (B) – 1.5

[The probability of an event E is a number such that .]

5. If , what is the probability of ‘ not E’ ?

Solution : We have ,

6. 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 ?

Solution : Total number of possible outcome

(i) The number of an orange flavoured candy in the bag

P(getting an orange flavoured candy)

(ii) The number of a lemon flavoured candy in the bag

P(getting a lemon flavoured candy)

7. 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 ?

Solution : Here,

We have ,

Therefore, the probability that the 2 students have the same birthday is 0.008 .

8. 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 ?

Solution : Total number of balls = 3 + 5 = 8

(i) The number of red balls = 3 .

P( getting red balls)

(ii) The number of balls are not red = 8 – 3 = 5 .

P(getting not red balls)

9. 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 probability that the marble taken out will be

(i) red ? (ii) white ? (iii) not green ?

Solution : Total number of balls = 5 + 8 + 4 = 17

(i) The number of red balls = 5 .

P( getting the red balls)

(ii) The number of white balls = 8 .

P( getting the white balls)

(iii) The number of balls are not green = 17 – 4 = 13 .

P(getting not red balls)

10. A piggy bank contains hundred 50p coins , fifty Rs 1 coins , twenty Rs 2 coins and ten Rs 5 coins . If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin :

(i) will be a 50p coin ?

(ii) will not be a Rs 5 coin ?

Solution : Total number of coins

(i) The number of a 50 p coins

P(getting a 50 p coin)

(ii) The number of coin will not be a Rs 5 coin

P( getting not be a Rs 5 coin)

11. 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 Fig. 15.4) . What is the probability that the fish taken out is a male fish ?

Solution : Total number of fish =

The number of male fish

P(getting a male fish)

12. 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 (see Fig. 15.5) and these are equally likely outcomes . 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 ?

Solution : Total number of possible outcome = 8

(i) The favourable number of elementary events = 1 [ i.e., 8 ]

P(getting 8)

(ii) The favourable number of elementary events = 4 [1 , 3 , 5 , 7 ]

P(getting an odd number)

(iii) The favourable number of elementary events = 6 [ 3 , 4 , 5 , 6 , 7 , 8 ]

P(getting an odd number)

(iv) The favourable number of elementary events = 8 [1 , 2, 3 , 4 , 5 , 6 , 7 , 8 ]

P(getting a number less than 9)

13. 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 .

Solution : Total number of possible outcome = 6

(i) The favourable number of elementary events = 3 [ 2 , 3 , 5 ]

P(getting a prime number)

(ii) The favourable number of elementary events = 3 [3 , 4 , 5 ]

P(getting a number lying between 2 and 6)

(iii) The favourable number of elementary events = 3 [ 1 , 3 , 5 ]

P(getting an odd number)

14. 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

Solution : Total number possible outcome .

(i) The number of a king of red colour

P(getting a king of red colour)

(ii) The number of a face card

P(getting a king of red colour)

(iii) The number of a red face card

P(getting a red face card)

(iv) The number of the jack of hearts

P(getting the jack of hearts)

(v) The number of a spade

P(getting a spade)

(vi) The number of the queen of diamonds

P(getting the queen of diamonds)

15. 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 ?

Solution: Total number of possible outcome

(i) The number of favourable outcome

P(getting the queen)

(ii) Total number of possible outcome [Queen out]

(a) P(getting an ace)

(b) P(getting a queen)

16. 12 defective pens are accidentally mixed with 132 good ones . It is not possible to just look at a pen and tell whethe or not it is defective . One pen is taken out at random from this lot . Determine the probability that the pen taken out is a good one .

Solution : Total number of pen

The number of good pen .

P(getting the pen taken out is a good one)

17. (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 ?

Solution : (i) Total number of bulbs .

The number of defective bulbs .

P(getting the bulbs is not defective)

(ii) Total number of bulbs [not replaced]

The number of the bulbs is not defective

P(getting the bulb is not defective)

18. 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 .

Solution : Total number of discs in the box

(i) The number of a two-digit number in box .

P(getting a two-digit number)

(ii) The number of a perfect square number in box

P(getting a perfect square number)

(iii) The number of discs divisible by

P(getting a number divisible by 5)

19. 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

Solution : Total number of possible outcome .

(i) The number of favourable outcome

P(getting A)

(ii) The number of favourable outcome

P(getting D)

20. Suppose you drop a die random on the rectangular region shown in Fig. 15.6 . What is the probability that it will land inside the circle with diameter 1 m ?

Solution : The total area of the rectangular region

Area of the circle

[Radius =12 m ]

P(getting the circle)

21. 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 ?

Solution: Total number of ball pens = 144 .

The number of defective ball pens = 20 .

The number of good ball pens = 144 – 20 =124

(i) P( getting a good ball pens) .

(ii) P(getting a defective ball pens)

22. Refer to Example 13. (i) complete the following table :

Event : ‘Sum on 2 dice’	Probability
2
3
4
5
6
7
8
9
10
11
12

(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 . Do you agree with this argument ? justify your answer .

Solution : The sample space {(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) , (6,1)

, (6,2) , (6,3) , (6,4) , (6,5) , (6,6)}

(i) We have ,

Event : ‘Sum on 2 dice’	Probability
2
3
4
5
6
7
8
9
10
11
12

(ii) No . because, the eleven sums are not equally likely .

23. 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 .

Solution: The sample space { HHH , HHT , HTT , TTH , THH , THT , HTH , TTT }

Total number of possible outcome .

The number of favourable outcome

P(Hanif will lose the game )

24. 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 ?

[Hint : Throwing a die twice and throwing two dice simultaneously are treated as the same experiment ]

Solution: The sample space

{(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) , (6,1) , (6,2) , (6,3) , (6,4) , (6,5) , (6,6)}

Total number of possible outcome

(i) The number of favourable outcome

P(not 5)

(ii) The number of favourable outcome

P(at least 5 once)

25. 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 .

(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 .

Solution : (i) Total number of possible outcome

The favourable number of outcome

P(getting one outcome)

Therefore, the arguments is not correct .

(ii) Total number of possible outcome

Total number of odd number

P(getting an odd number)

Therefore, the arguments is correct .

**Class 10 Maths Chapter 15. PROBABILITY EXERCISE 15.2 (Optional)* Solutions**

1. 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?
2. A die is numbered in such a way that its faces show the numbers 1, 2, 2, 3, 3, 6. It is thrown two times and the total score in two throws is noted. Complete the following table which gives a few values of the total score on the two throws:

+	1 2 2 3 3 6
1	2 3 3 4 4 7
2	3 4 4 5 5 8
2	5
3
3	5 9
6	7 8 8 9 9 12

What is the probability that the total score is (i) even? (ii) 6? (iii) at least 6?
3. 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.
4. 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 .

Solution:

5. 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 ⋅ Find the number of blue balls in the jar.