## NCERT Solution for Class 10 Maths Chapter 14 Probability

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**Question.1. Complete the following statements:**

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

**Answer.** 1

**(ii) The probability of an event that cannot happen is ______. Such an event is called ________.**

**Answer.** 0, impossible event

**(iii) The probability of an event that is certain to happen is _______. Such an event is called ________.**

**Answer. **1, sure event or certain event

**(iv) The sum of the probabilities of all the elementary events of an experiment is ______.**

**Answer.** 1

**(v) The probability of an event is greater than or equal to _______ and less than or equal to ________.**

**Answer.** 0, 1

**Question.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.**

**Answer.** It is not an equally likely event, as it depends on various factors such as whether the car will start or not. And factors for both the conditions are not the same.

**(ii) A player attempts to shoot a basketball. She/he shoots or misses the shot.**

**Answer.** It is not an equally likely event, as it depends on the player’s ability and there is no information given about that.

**(iii) A trial is made to answer a true-false question. The answer is right or wrong.**

**Answer.** It is an equally likely event.

**(iv) A baby is born. It is a boy or a girl.**

**Answer.** It is an equally likely event.

**Question.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?**

**Answer.** When we toss a coin, the possible outcomes are only two, head or tail, which are equally likely outcomes. Therefore, the result of an individual toss is completely unpredictable.

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

**(A) \frac{2}{3} **

**(B) -1.5**

**(C) 15%**

**(D) 0.7**

**Answer.** Probability of an event (E) is always greater than or equal to 0. Also, it is always less than or equal to one. This implies that the probability of an event cannot be negative or greater than 1. Therefore, out of these alternatives, −1.5 cannot be a probability of an event.

Hence, (B) is the correct option.

**Question.5. If P(E) = 0.05, what is the probability of ‘not E’?**

**Answer.** We know that,

P (\overline{E}) = 1-P(E)

⇒ P (\overline{E})= 1-0.05 = 0.95

Therefore, the probability of ‘not E’ is 0.95.

**Question.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?**

**Answer.** The bag contains lemon flavoured candies only. It does not contain any orange flavoured candies. This implies that every time, she will take out only lemon flavoured candies. Therefore, event that Malini will take out an orange flavoured candy is an impossible event.

Hence, P (an orange flavoured candy) = 0

**(ii) a lemon flavoured candy?**

**Answer.** As the bag has lemon flavoured candies, Malini will take out only lemon flavoured candies. Therefore, event that Malini will take out a lemon flavoured candy is a sure event.

P (a lemon flavoured candy) = 1

**Question.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?**

**Answer.** Probability that two students are not having same birthday P(\overline{E})= 0.992

Probability that two students are having same birthday P(E)=1- P(\overline{E})

⇒ P(E)=1- 0.992

⇒ P(E)= 0.008

**Question.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?**

**Answer. **Total number of balls in the bag = 8

Probability of getting a red ball = \frac{Number of favourable outcomes}{Number of total outcomes}

Probability of getting a red ball = \frac{3}{8}

**(ii) not red?**

**Answer.** Probability of not getting red ball = 1 − Probability of getting a red ball

⇒ Probability of not getting red ball = 1- \frac{3}{8}

⇒ Probability of not getting red ball = \frac{5}{8}

**Question.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 the probability that the marble taken out will be**

**(i) red?**

**Answer.** Total number of marbles = 5 + 8 + 4 = 17

Number of red marbles = 5

Probability of getting a red marble = \frac{Number of favourable outcomes}{Number of total outcomes}

Probability of getting a red marble = \frac{5}{17}

**(ii) white?**

**Answer.** Number of white marbles = 8

Probability of getting a white marble = \frac{Number of favourable outcomes}{Number of total outcomes}

Probability of getting a white marble = \frac{8}{17}

**(iii) not green?**

**Answer.** Number of green marbles = 4

Probability of getting a green marble = \frac{Number of favourable outcomes}{Number of total outcomes}

Probability of getting a green marble = \frac{4}{17}

Possibility of not getting a green marble = 1-\frac{4}{17} = \frac{13}{17}

**Question.10. A piggy bank contains hundred 50 p coins, fifty ₹ 1 coins, twenty ₹ 2 coins and ten ₹ 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 50 p coin?**

**Answer.** Total number of coins in a piggy bank = 100 + 50 + 20 + 10 = 180

Number of 50 p coins = 100

Probability of getting a 50 p coin = \frac{Number of favourable outcomes}{Number of total outcomes}

Probability of getting a 50 p coin = \frac{100}{180}= \frac{5}{9}

**(ii) Will not be a ₹ 5 coin?**

**Answer.** Number of ₹ 5 coins = 10

Probability of getting a ₹ 5 coin = \frac{Number of favourable outcomes}{Number of total outcomes}

Probability of getting a ₹ 5 coin = \frac{10}{180} = \frac{1}{18}

Probability of not getting a Rs 5 coin = 1-\frac{1}{18} = \frac{17}{18}

**Question.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 the given figure). What is the probability that the fish taken out is a male fish?**

**Answer. **Total number of fishes in a tank = (Number of male fishes + Number of female fishes)

Total number of fishes in a tank = 5 + 8 = 13

Probability of getting a male fish = \frac{Number of favourable outcomes}{Number of total possible outcomes}

Probability of getting a male fish = \frac{5}{13}

**Question.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 the given figure), and these are equally likely outcomes. What is the probability that it will point at

**(i) 8?**

**Answer.** Total number of possible outcomes = 8

Probability of getting 8 = \frac{Number of favourable outcomes}{Number of total possible outcomes}

Probability of getting 8 = \frac{1}{8}

**(ii) an odd number?**

**Answer.** Total number of odd numbers on spinner = 4

Probability of getting an odd number = \frac{Number of favourable outcomes}{Number of total possible outcomes}

Probability of getting an odd number = \frac{4}{8} = \frac{1}{2}

**(iii) a number greater than 2?**

**Answer. **The numbers greater than 2 are 3, 4, 5, 6, 7, and 8. Therefore, total numbers greater than 2 = 6

Probability of getting a number greater than 2 = \frac{Number of favourable outcomes}{Number of total possible outcomes}

Probability of getting a number greater than 2 = \frac{6}{8} = \frac{3}{4}

**(iv) a number less than 9?**

**Answer.** The numbers less than 9 are 1, 2, 3, 4, 6, 7, and 8.

Therefore, total numbers less than 9 = 8

Probability of getting a number less than 9 = \frac{8}{8}=1

**Question.13. A die is thrown once. Find the probability of getting**

**(i) a prime number;**

**Answer.** The possible outcomes when a dice is thrown = {1, 2, 3, 4, 5, 6}

Number of possible outcomes of a dice = 6

Prime numbers on a dice are 2, 3, and 5.

Total prime numbers on a dice = 3

Probability of getting a prime number = \frac{3}{6} = \frac{1}{2}

**(ii) a number lying between 2 and 6;**

**Answer.** Numbers lying between 2 and 6 = 3, 4, 5

Total numbers lying between 2 and 6 = 3

Probability of getting a number lying between 2 and 6 = \frac{3}{6} = \frac{1}{2}

**(iii) an odd number.**

**Answer.** Odd numbers on a dice = 1, 3, and 5

Total odd numbers on a dice = 3

Probability of getting an odd number = \frac{3}{6} = \frac{1}{2}

**Question.14. One card is drawn from a well-shuffled deck of 52 cards. Find the probability of getting**

**(i) a king of red colour**

**Answer.** Total number of cards in a well-shuffled deck = 52

Total number of kings of red colour = 2

P (getting a king of red colour) =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting a king of red colour) =\frac{2}{52}=\frac{1}{26}

**(ii) a face card**

**Answer.** Total number of face cards = 12

P (getting a face card) =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting a face card) =\frac{12}{52}=\frac{3}{13}

**(iii) a red face card**

**Answer.** Total number of red face cards = 6

P (getting a red face card) =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting a red face card) =\frac{6}{52}=\frac{3}{26}

**(iv) the jack of hearts**

**Answer. **Total number of Jack of hearts = 1

P (getting a Jack of hearts) =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting a Jack of hearts) =\frac{1}{52}

**(v) a spade**

**Answer. **Total number of spade cards = 13

P (getting a spade card) =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting a spade card) =\frac{13}{52}=\frac{1}{4}

**(vi) the queen of diamonds**

**Answer.** Total number of queen of diamonds = 1

P (getting a queen of diamond) =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting a queen of diamond) =\frac{1}{52}

**Question.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?**

**Answer.** Total number of cards = 5

Total number of queens = 1

P (getting a queen) =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting a queen) =\frac{1}{5}

**(ii) If the queen is drawn and put aside, what is the probability that the second card picked up is**

**(a) an ace?**

**Answer.** When the queen is drawn and put aside, the total number of remaining cards will be 4.

Total number of aces = 1

P (getting an ace) =\frac{1}{4}

**(b) a queen?**

**Answer.** As queen is already drawn, therefore, the number of queens will be 0.

P (getting a queen) =\frac{0}{4}=0

**Question.16. 12 defective pens are accidentally mixed with 132 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.**

**Answer.** Total number of pens = 12 + 132 = 144

Total number of good pens = 132

P (getting a good pen) =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting a good pen) =\frac{132}{144}=\frac{11}{12}

**Question.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?**

**Answer.** Total number of bulbs = 20

Total number of defective bulbs = 4

P (getting a defective bulb) =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting a defective bulb) =\frac{4}{20}=\frac{1}{5}

**(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?**

**Answer. **Remaining total number of bulbs = 19

Remaining total number of non-defective bulbs = 16 − 1 = 15

P (getting a not defective bulb) =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting a not defective bulb) =\frac{15}{19}

**Question.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**

**Answer. **Total number of discs = 90

Total number of two-digit numbers between 1 and 90 = 81

P (getting a two-digit number) =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting a two-digit number) =\frac{81}{90}=\frac{9}{10}

**(ii) a perfect square number**

**Answer.** Perfect squares between 1 and 90 are 1, 4, 9, 16, 25, 36, 49, 64, and 81.

Therefore, total number of perfect squares between 1 and 90 is 9.

P (getting a perfect square) =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting a perfect square) =\frac{9}{90}=\frac{1}{10}

**(iii) a number divisible by 5.**

**Answer.** Numbers that are between 1 and 90 and divisible by 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, and 90. Therefore, total numbers divisible by 5 = 18

Probability of getting a number divisible by 5 =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting a perfect square) =\frac{18}{90}=\frac{1}{5}

**Question.19.** **A child has a die whose six faces shows the letters as given below:**

**The die is thrown once. What is the probability of getting**

**(i) A?**

**Answer.** Total number of possible outcomes on the dice = 6

(i) Total number of faces having A on it = 2

P (getting A) =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting A) =\frac{2}{6}=\frac{1}{3}

**(ii) D?**

**Answer.** Total number of faces having D on it = 1

P (getting D) =\frac{Number of favourable outcomes}{Number of total possible outcomes}

P (getting a getting D) =\frac{1}{6}

**Question.20.** **Suppose you drop a die at random on the rectangular region shown in the given figure.**

**What is the probability that it will land inside the circle with diameter 1 m?**

**Answer.** Area of rectangle =l\times b = 3\times 2 = 6 m^{2}

Area of circle (of diameter 1 m) =\pi r^{2} = \pi \left(\frac{1}{2}\right)^{2}=\frac{\pi}{4}

P (die will land inside the circle) =\frac{\left(\frac{\pi}{4}\right)}{6}=\frac{\pi}{24}

**Question.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?**

**Answer.** Total number of pens = 144

Total number of defective pens = 20

Total number of good pens = 144 − 20 = 124

Probability of getting a good pen =\frac{Number of favourable outcomes}{Number of total possible outcomes}

Probability of getting a good pen =\frac{124}{144}=\frac{31}{36}

P (Nuri buys a pen) =\frac{31}{36}

**(ii) She will not buy it?**

**Answer.** P (Nuri will not buy a pen) =1-\frac{31}{36}=\frac{5}{36}

**Question.22. Two dice, one blue and one grey, are thrown at the same time.**

**(i) Write down all the possible outcomes and complete the following table**

**Answer.** It can be observed that,

To get the sum as 2, possible outcomes = (1, 1)

To get the sum as 3, possible outcomes = (2, 1) and (1, 2)

To get the sum as 4, possible outcomes = (3, 1), (1, 3), (2, 2)

To get the sum as 5, possible outcomes = (4, 1), (1, 4), (2, 3), (3, 2)

To get the sum as 6, possible outcomes = (5, 1), (1, 5), (2, 4), (4, 2), (3, 3)

To get the sum as 7, possible outcomes = (6, 1), (1, 6), (2, 5), (5, 2), (3, 4), (4, 3)

To get the sum as 8, possible outcomes = (6, 2), (2, 6), (3, 5), (5, 3), (4, 4)

To get the sum as 9, possible outcomes = (3, 6), (6, 3), (4, 5), (5, 4)

To get the sum as 10, possible outcomes = (4, 6), (6, 4), (5, 5)

To get the sum as 11, possible outcomes = (5, 6), (6, 5)

To get the sum as 12, possible outcomes = (6, 6)

**(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 1/11. Do you agree with this argument?**

**Answer.** Probability of each of these sums will not be =\frac{1}{11} as these sums are not equally likely.

**Question.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.**

**Answer.** The possible outcomes are

{HHH, TTT, HHT, HTH, THH, TTH, THT, HTT}

Number of total possible outcomes = 8

Number of favourable outcomes = 2 {i.e., TTT and HHH}

P (Hanif will win the game) =\frac{2}{8}=\frac{1}{4}

P (Hanif will lose the game) =1-\frac{1}{4}=\frac{3}{4}

**Question.24. A die is thrown twice. What is the probability that**

**(i) 5 will not come up either time?**

**Answer.** Total number of outcomes =6\times 6=36

Total number of outcomes when 5 comes up on either time are (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (1, 5), (2, 5), (3, 5), (4, 5), (6, 5)

Hence, total number of favourable cases = 11

P (5 will come up either time) =\frac{11}{36}

P (5 will not come up either time) =1-\frac{11}{36}=\frac{25}{36}

**(ii) 5 will come up at least once?**

**Answer.** Total number of cases, when 5 can come at least once = 11

P (5 will come at least once) =\frac{11}{36}

**Question.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 \frac{1}{3}.**

**Answer.** Incorrect

When two coins are tossed, the possible outcomes are (H, H), (H, T), (T, H), and (T, T). It can be observed that there can be one of each in two possible ways − (H, T), (T, H).

Therefore, the probability of getting two heads is \frac{1}{4}, the probability of getting two tails is \frac{1}{4}, and the probability of getting one of each is \frac{1}{2}.

It can be observed that for each outcome, the probability is not \frac{1}{3}.

**(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 \frac{1}{2}.**

**Answer.** Correct

When a dice is thrown, the possible outcomes are 1, 2, 3, 4, 5, and 6. Out of these, 1, 3, 5 are odd and 2, 4, 6 are even numbers.

Therefore, the probability of getting an odd number is \frac{1}{2}.