Chapter 14 · Probability

NCERT Class 10 Maths Exercise 14.1

All 25 questions of NCERT Class 10 Maths Exercise 14.1 (Probability), in the book's order, each with its answer and a step-by-step solution.

1. Q1
   Complete the following statements:

   1. (i)Probability of an event E + Probability of the event 'not E' = ____.
   2. (ii)​The probability of an ​event that cannot ​happen is ____. Such an event is called ____.
   3. (iii)The probability of an event that is certain to happen is ____. Such an event is called ____.
   4. (iv)The sum ​of the probabilities of all the elementary events of an ​experiment is ____.
   5. (v)The probability of an event is greater ​than or equal to ____ and ​less than or equal ​to ____.
   Show answer

   (i) $1$; (ii) $0$, an impossible event; (iii) $1$, a sure (certain) event; (iv) $1$; (v) $0$ and $1$.

   Step-by-step solution →
2. Q2
   Which of the following ​experiments ​have equally likely outcomes? ​Explain.

   1. (i)A driver attempts to start a car. ​The car starts ​or does not start.
   2. (ii)A player attempts to shoot a basketball. She/he shoots or misses the shot.
   3. (iii)A trial is made to answer a true-false question. The answer is right or ​wrong.
   4. (iv)A baby is born. ​It is a boy or a girl.
   Show answer

   (i) Not equally likely — the car starting depends on its condition. (ii) Not equally likely — shooting depends on the player's skill. (iii) Equally likely — a random guess favours neither answer. (iv) Equally likely — a boy and a girl have no built-in bias.

   Step-by-step solution →
3. Q3
   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?
   Show answer

   Tossing an unbiased coin gives two equally-likely outcomes, so $P(\text{head})=P(\text{tail})=\frac{1}{2}$. Each team is tied to one outcome, hence each has an equal $\frac{1}{2}$ chance and no team is favoured — that is why the toss is fair.

   Step-by-step solution →
4. Q4
   Which ​of the ​following cannot ​be ​the ​probability of ​an event?

   1. (a)$\frac{2}{3}$
   2. (b)$-1.5$
   3. (c)$15\%$
   4. (d)$0.7$
   Show answer

   (b) $-1.5$

   Step-by-step solution →
5. Q5
   If $P(E)=0.05$, what is ​the ​probability ​of ​'not ​E'?
   Show answer

   $P(\text{not }E)=0.95$.

   Step-by-step solution →
6. Q6
   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 ​

   1. (i)an orange flavoured candy?
   2. (ii)a lemon flavoured candy?
   Show answer

   (i) $P(\text{orange})=0$ (impossible event); (ii) $P(\text{lemon})=1$ (sure event).

   Step-by-step solution →
7. Q7
   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?
   Show answer

   $P(\text{same birthday})=1-0.992=0.008$.

   Step-by-step solution →
8. Q8
   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

   1. (i)​red?
   2. (ii)not red?
   Show answer

   (i) $P(\text{red})=\frac{3}{8}$; (ii) $P(\text{not red})=\frac{5}{8}$.

   Step-by-step solution →
9. Q9
   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 ​

   1. (i)red?
   2. (ii)white?
   3. (iii)not ​green?
   Show answer

   (i) $P(\text{red})=\frac{5}{17}$; (ii) $P(\text{white})=\frac{8}{17}$; (iii) $P(\text{not green})=\frac{13}{17}$.

   Step-by-step solution →
10. Q10
    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

    1. (i)will be a 50 p coin?
    2. (ii)will ​not be ​a Rs 5 coin?
    Show answer

    (i) $P(50\text{p})=\frac{5}{9}$; (ii) $P(\text{not Rs}5)=\frac{17}{18}$.

    Step-by-step solution →
11. Q11
    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. 14.4). ​What is ​the probability that the fish taken out is a ​male ​fish?

    Show answer

    $P(\text{male fish})=\frac{5}{13}$.

    Step-by-step solution →
12. Q12
    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. ​14.5), and ​these are equally likely outcomes. What is the probability that it will point ​at

    1. (i)$8$? ​
    2. (ii)an ​odd number?
    3. (iii)a number greater than $2$?
    4. (iv)a number less than $9$?

    Show answer

    (i) $\frac{1}{8}$; (ii) $\frac{1}{2}$; (iii) $\frac{3}{4}$; (iv) $1$.

    Step-by-step solution →
13. Q13
    A die is thrown ​once. Find the ​probability of getting

    1. (i)a prime ​number;
    2. (ii)​a number ​lying between 2 ​and 6;
    3. (iii)an odd number.
    Show answer

    (i) $P(\text{prime})=\frac{1}{2}$; (ii) $P(\text{between }2\text{ and }6)=\frac{1}{2}$; (iii) $P(\text{odd})=\frac{1}{2}$.

    Step-by-step solution →
14. Q14
    One card is drawn ​from a well-shuffled ​deck of ​52 cards. ​Find the probability of getting

    1. (i)a king of red colour
    2. (ii)a face card
    3. (iii)a red ​face ​card
    4. (iv)​the jack of hearts
    5. (v)a spade ​
    6. (vi)the queen of diamonds
    Show answer

    (i) $\frac{1}{26}$; (ii) $\frac{3}{13}$; (iii) $\frac{3}{26}$; (iv) $\frac{1}{52}$; (v) $\frac{1}{4}$; (vi) $\frac{1}{52}$.

    Step-by-step solution →
15. Q15
    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.

    1. (i)What ​is the probability ​that the card is the queen? ​
    2. (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?
    Show answer

    (i) $P(\text{queen})=\frac{1}{5}$; (ii)(a) $P(\text{ace})=\frac{1}{4}$; (ii)(b) $P(\text{queen})=0$.

    Step-by-step solution →
16. Q16
    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.
    Show answer

    $P(\text{good pen})=\frac{11}{12}$.

    Step-by-step solution →
17. Q17
    1. (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?
    2. (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?
    Show answer

    (i) $P(\text{defective})=\frac{1}{5}$; (ii) $P(\text{not defective})=\frac{15}{19}$.

    Step-by-step solution →
18. Q18
    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

    1. (i)a two-digit number,
    2. (ii)a ​perfect square number,
    3. (iii)a number ​divisible by $5$.
    Show answer

    (i) $P(\text{two-digit})=\frac{9}{10}$; (ii) $P(\text{perfect square})=\frac{1}{10}$; (iii) $P(\text{divisible by }5)=\frac{1}{5}$.

    Step-by-step solution →
19. Q19
    A child has a ​die whose six ​faces show the letters ​A, ​B, C, D, ​E, ​A. The die is thrown once. ​What is the probability of getting ​

    1. (i)$A$?
    2. (ii)$D$?
    Show answer

    (i) $P(A)=\frac{1}{3}$; (ii) $P(D)=\frac{1}{6}$.

    Step-by-step solution →
20. Q20
    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 $1\text{m}$?

    Show answer

    $P(\text{lands inside the circle})=\frac{\pi }{24}\approx 0.131$.

    Step-by-step solution →
21. Q21
    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

    1. (i)She will buy it?
    2. (ii)She will not ​buy it?
    Show answer

    (i) $P(\text{she will buy})=\frac{31}{36}$; (ii) $P(\text{she will not buy})=\frac{5}{36}$.

    Step-by-step solution →
22. Q22
    Refer ​to Example ​13.

    1. (i)Complete ​the following table giving the probability ​for each possible sum when two dice are ​thrown together. Some entries are filled in to help you.
    2. (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 $\frac{1}{11}$. Do you agree with this argument? Justify ​your answer.

    Sum on 2 dice and its probability (entries to be completed in part (i))

    Event: Sum on 2 dice	2	3	4	5	6	7	8	9	10	11	12
    Probability	$\frac{1}{36}$						$\frac{5}{36}$				$\frac{1}{36}$

    Show answer

    (i) $P(\text{sum})$ for $2,3,\dots ,12$ is $\frac{1}{36},\frac{2}{36},\frac{3}{36},\frac{4}{36},\frac{5}{36},\frac{6}{36},\frac{5}{36},\frac{4}{36},\frac{3}{36},\frac{2}{36},\frac{1}{36}$. (ii) No. The $11$ sums are not equally likely (e.g. $P(7)=\frac{6}{36}$ but $P(2)=\frac{1}{36}$), so each is not $\frac{1}{11}$.

    Step-by-step solution →
23. Q23
    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.
    Show answer

    $P(\text{Hanif loses})=\frac{3}{4}=0.75$.

    Step-by-step solution →
24. Q24
    A die is thrown ​twice. What is the probability that

    1. (i)$5$ ​will ​not come ​up either time?
    2. (ii)$5$ ​will come up at least once? ​[Hint: Throwing a die twice and throwing two dice simultaneously are treated as ​the same experiment.]
    Show answer

    (i) $P(\text{no }5)=\frac{25}{36}$; (ii) $P(\text{at least one }5)=\frac{11}{36}$.

    Step-by-step solution →
25. Q25
    Which of the following arguments are correct and which are ​not correct? Give ​reasons for your answer.

    1. (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 $\frac{1}{3}$.
    2. (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 $\frac{1}{2}$.
    Show answer

    (i) Not correct — the three outcomes are not equally likely ('one of each' covers HT and TH, so $P=\frac{1}{4},\frac{1}{4},\frac{1}{2}$, not $\frac{1}{3}$ each). (ii) Correct — three of the six faces are odd, so $P(\text{odd})=\frac{1}{2}$.

    Step-by-step solution →