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YOUR STUDY SPACE · CBSE

Probability — Class 10 Mathematics

Mathematics · Chapter 14

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Chapter summary

Probability measures how likely an event is to happen, on a scale from 0 (impossible) to 1 (certain). This chapter develops the theoretical (classical) definition: the probability of an event equals the number of favourable outcomes divided by the total number of equally likely outcomes. It applies this to everyday experiments such as tossing coins, rolling dice and drawing cards, and uses the fact that the probabilities of an event and its complement add up to 1. Probability provides a precise language for reasoning about chance and uncertainty.
Theoretical probabilitySimple experiments (coins, dice, cards)Complementary events

Key ideas & terms

Probability of an event
.
Sure/impossible event
Probability 1 (certain) or 0 (impossible).
Complementary event
.
Equally likely outcomes
Outcomes that each have the same chance of occurring.

Extra questions & answers

What is the probability of a sure event and an impossible event?

for a sure event and for an impossible event.

If , what is ?

.

Write the formula for the theoretical probability of an event E.

.

What is the probability of a sure event and of an impossible event?

for a sure (certain) event and for an impossible event.

What is the range of the probability of any event?

The probability of any event lies between 0 and 1, that is .

What are complementary events, and how are their probabilities related?

An event and "not " are complementary; , so .

A die is thrown once. What is the probability of getting an even number?

Favourable outcomes are 2, 4, 6 (three of them) out of six, so .

A card is drawn from a well-shuffled pack of 52. What is the probability that it is a king?

There are 4 kings, so .

Two coins are tossed together. What is the probability of getting two heads?

The outcomes are HH, HT, TH, TT; only one is favourable, so .

If the probability of an event is 0.7, what is the probability of its not happening?

.

EXERCISE 14.1

1. 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 ______.

(i) Probability of an event + Probability of the event ‘not ’ = .
(ii) The probability of an event that cannot happen is . Such an event is called an impossible event.
(iii) The probability of an event that is certain to happen is . Such an event is called a sure event or a certain event.
(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 .

2. 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.

(i) The outcomes are not necessarily equally likely because the chance of the car starting depends on its condition and other factors.
(ii) The outcomes are not necessarily equally likely because the chance of scoring depends on the player’s skill and the conditions of the shot.
(iii) The outcomes are not necessarily equally likely because the chance of a correct answer depends on the person’s knowledge.
(iv) The outcomes are treated as equally likely because the birth of a boy and the birth of a girl are assumed to have equal chances.

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?

A fair coin has two equally likely outcomes, head and tail. Therefore, when each team is assigned one outcome, both teams have the same probability, , of getting the ball.

4. Which of the following cannot be the probability of an event?
(A) (B) (C) (D)

The probability of an event must satisfy . Since , option (B), , cannot be the probability of an event.

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

Using ,

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?

(i) There are no orange flavoured candies, so the probability is .
(ii) Every candy is lemon flavoured, so the probability is .

7. It is given that in a group of students, the probability of students not having the same birthday is . What is the probability that the students have the same birthday?

The two events are complementary. Therefore,

8. A bag contains red balls and 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?

There are balls.
(i)
(ii) A ball that is not red is black, so

9. A box contains red marbles, white marbles and 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?

The total number of marbles is .
(i)
(ii)
(iii) There are marbles that are not green. Hence,

10. A piggy bank contains hundred coins, fifty coins, twenty coins and ten 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 coin ? (ii) will not be a coin?

The total number of coins is

(i)
(ii) The number of coins that are not coins is . Therefore,

11. Gopi buys a fish from a shop for his aquarium. The shopkeeper takes out one fish at random from a tank containing male fish and female fish (see Fig. 14.4). What is the probability that the fish taken out is a male fish?

The total number of fish is . Therefore,

12. A game of chance consists of spinning an arrow which comes to rest pointing at one of the numbers (see Fig. 14.5 ), and these are equally likely outcomes. What is the probability that it will point at
(i) ?
(ii) an odd number?
(iii) a number greater than ?
(iv) a number less than ?

There are equally likely outcomes.
(i) Only one outcome is , so

(ii) The odd numbers are . Hence,

(iii) The numbers greater than are . Thus,

(iv) All the given numbers are less than . Therefore,

13. A die is thrown once. Find the probability of getting
(i) a prime number; (ii) a number lying between and ; (iii) an odd number.

The possible outcomes are .
(i) The prime numbers are . Therefore,

(ii) The numbers lying between and are . Therefore,

(iii) The odd numbers are . Hence,

14. One card is drawn from a well-shuffled deck of 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

A deck has cards.
(i) There are red kings. Thus,

(ii) There are face cards. Thus,

(iii) There are red face cards. Thus,

(iv) There is one jack of hearts. Hence,

(v) There are spades. Hence,

(vi) There is one queen of diamonds. Hence,

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?

(i) There is one queen among cards. Therefore,

(ii) After the queen is removed, cards remain.
(a) There is one ace among them, so

(b) No queen remains, so

16. defective pens are accidentally mixed with good ones. It is not possible to just look at a pen and tell whether 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.

The total number of pens is . Therefore,

17. (i) A lot of bulbs contain 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 ?

(i) There are defective bulbs out of . Hence,

(ii) Initially there are non-defective bulbs. After one non-defective bulb is removed, non-defective bulbs remain among bulbs. Therefore,

18. A box contains discs which are numbered from to . 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 .

There are possible outcomes.
(i) The two-digit numbers are from to , giving numbers. Thus,

(ii) The perfect squares from to are , giving outcomes. Hence,

(iii) There are multiples of from to . Therefore,

19. A child has a die whose six faces show the letters as given below:
A B C D E A
The die is thrown once. What is the probability of getting (i) A? (ii) D?

The die has equally likely faces.
(i) The letter A appears on faces. Therefore,

(ii) The letter D appears on one face. Therefore,

20*. Suppose you drop a die at random on the rectangular region shown in Fig. 14.6. What is the probability that it will land inside the circle with diameter ?

The rectangle has dimensions by , so its area is

The circle has radius , so its area is

Therefore,

21. A lot consists of ball pens of which 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 ?

The number of good pens is .
(i) Nuri will buy a good pen. Hence,

(ii) She will not buy a defective pen. Hence,

22. Refer to Example 13. (i) Complete the following table:
Event :
‘Sum on dice’
Probability ______ ______ ______ ______ ______ ______ ______ ______
(ii) A student argues that ‘there are possible outcomes and . Therefore, each of them has a probability . Do you agree with this argument? Justify your answer.

(i) The completed probabilities for sums through , respectively, are

(ii) No. The sums are not equally likely. For example, a sum of occurs only with , whereas a sum of occurs with . Thus, their probabilities are and , respectively, rather than each.

23. A game consists of tossing a one rupee coin 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.

The equally likely outcomes are . Hanif wins for and , so he loses for the remaining outcomes. Therefore,

24. A die is thrown twice. What is the probability that
(i) will not come up either time? (ii) will come up at least once?
[Hint : Throwing a die twice and throwing two dice simultaneously are treated as the same experiment]

There are equally likely outcomes.
(i) On each throw, there are outcomes other than . Thus, there are outcomes in which does not appear. Therefore,

(ii) This event is complementary to the event in part (i). Hence,

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 .

(i) The argument is not correct. The four equally likely elementary outcomes are . Two heads has probability , two tails has probability , and one of each has probability . Therefore, the three stated outcomes are not equally likely.
(ii) The argument is correct. A die has three odd outcomes, , and three even outcomes, . Hence,