Exercise 15.1
Q.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 as event is called _________.
(iii) The probability of an event that is certain to happen is _________. Such as 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 _______.
Sol.1) (i) 1
(ii) 0, impossible event
(iii) 1, sure event or certain event
(iv) 1
(v) 0, 1
Q.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.
Sol.2) (i) 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) It is not an equally likely event, as it depends on the player’s ability and there is no information given about that.
(iii) It is an equally likely event.
(iv) It is an equally likely event.
Q.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?
Sol.3) 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.
Q.4) Which of the following cannot be the probability of an event?
i) 2/3
ii) −1.5 iii) 15% iv) 0.7
Sol.4) 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)
Q.5) If P(E) = 0.05, what is the probability of ‘not E’?
Sol.5) We know that,
𝑃(𝐸 ̅ ) = 1 − 𝑃(𝐸)
𝑃(𝐸 ̅ ) = 1 − 0.05 = 0.95
Therefore, the probability of ‘not E’ is 0.95.
Q.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?
Sol.6) (i) 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) 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
Q.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?
Sol.7) Probability that two students are not having same birthday 𝑃(𝐸 ̅) = 0.992
Probability that two students are having same birthday 𝑃 (𝐸) = 1 − 𝑃(𝐸 ̅ )
= 1 − 0.992
= 0.008
Q.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?
Sol.8) (i) Total number of balls in the bag = 8
(ii) Probability of not getting red ball = 1 − Probability of getting a red ball
= 1 − 3/8 = 5/8
Q.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? (ii) white? (iii) not green?
Sol.9) Total number of marbles = 5 + 8 + 4 = 17
(i) Number of red marbles = 5
Q.10) A piggy bank contains hundred 50 p 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 50 p coin?
(ii) Will not be a Rs.5 coin?
Sol.10) Total number of coins in a piggy bank = 100 + 50 + 20 + 10 = 180
(i) Number of 50 p coins = 100
Q.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?
Sol.11) Total number of fishes in a tank
= Number of male fishes + Number of female fishes
= 5 + 8 = 13
Probability of getting a male fish = 𝑁𝑜.𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 / 𝑁𝑜.𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 = 5/13
Q.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? (ii) an odd number?
(iii) a number greater than 2?
(iv) a number less than 9?
Sol.12) (i) To land on the number 8, a singular fate,
In the circle of choices, a rendezvous with eight.
The probability, a simple quest,
Is 1 out of 8, a fraction at best.
(ii) Now, let's embrace the realm of odd,
Numbers dancing, a whimsical squad.
From 1 to 8, the odds do sway,
4 out of 8, the arrow's array.
(iii) Beyond the realm of mere 2,
A quest for numbers greater, anew.
From 3 to 8, the arrow's flight,
6 out of 8, a probability bright.
(iv) To dwell in the kingdom of less than 9,
A realm where all numbers shine.
From 1 to 8, each a familiar sign,
The probability, a perfect design.
Q.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.
Sol.13) The possible outcomes when a dice is thrown = {1, 2, 3, 4, 5, 6}
Number of possible outcomes of a dice = 6
(i) Prime numbers on a dice are 2, 3, and 5.
Total prime numbers on a dice = 3
Probability of getting a prime number = 3/9 = 1/2
(ii) 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 = 3/6 = 1/2
(iii) Odd numbers on a dice = 1, 3, and 5
Total odd numbers on a dice = 3
Probability of getting an odd number= 3/6 = 1/2
Q.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
Sol.14) Total number of cards in a well-shuffled deck = 52
(i) Total number of kings of red colour = 2
Q.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?
Sol.15) (i) Total number of cards = 5
Total number of queens = 1
P (getting a queen) = 𝑁𝑜.𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 / 𝑁𝑜.𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 = 1/5
(ii) When the queen is drawn and put aside, the total number of remaining cards will be 4.
(a) Total number of aces = 1/P (getting an ace)= 1/4
(b) As queen is already drawn, therefore, the number of queens will be 0.
P (getting a queen) = 0/4 = 0
Q.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.
Sol.16) Total number of pens = 12 + 132 = 144
Total number of good pens = 132
P (getting a good pen) = 𝑁𝑜.𝑜𝑓 𝑓𝑎𝑣𝑜𝑢𝑟𝑎𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 / 𝑁𝑜.𝑜𝑓 𝑡𝑜𝑡𝑎𝑙 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠 = 132/144 = 11/12
Q.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?
Sol.17) (i) Total number of bulbs = 20
Total number of defective bulbs = 4
Q.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.
Sol.18) Total number of discs = 90
(i) Total number of two-digit numbers between 1 and 90 = 81
P (getting a two-digit number)= 81/90 = 9/10
(ii) 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)= 9/90 = 1/10
(iii) 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= 18/90 = 1/5
Q.19) A child has a die whose six faces shows 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?
Sol.19) Total number of possible outcomes on the dice = 6
(i) Total number of faces having A on it = 2
P (getting A) = 2/6 = 1/3
(ii) Total number of faces having D on it = 1
P (getting D)= 1/6
Q.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?
Sol.20) Area of rectangle = 𝑙 × 𝑏 = 3 × 2 = 6 𝑚2
Q.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?
Sol.21) Total number of pens = 144
Total number of defective pens = 20
Total number of good pens = 144 − 20 = 124
(i) Probability of getting a good pen = 124/144 = 31/36
P (Nuri buys a pen)= 31/36
(ii) P (Nuri will not buy a pen) = 1 − 31/36 = 5/36
Q.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:
(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?
Sol.22) (i) 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) Probability of each of these sums will not be 1/11 as these sums are not equally likely
Q.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.
Sol.23) 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) = 2/8 = 1/4
P (Hanif will lose the game) = 1 − 1/4 = 3/4
Q.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].
Sol.24) Total number of outcomes = 6 × 6 = 36
(i) 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) = 11/36
P (5 will not come up either time) = 1 − (11/36) = 25/36
(ii) Total number of cases, when 5 can come at least once = 11
P (5 will come at least once) = 11/36
Q.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 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 1/2.
Sol.25) (i) 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 1/4, the probability of getting two tails is 1/4, and the probability of getting one of each is 1/2. It can be observed that for each outcome, the probability is not 1/3.
(ii) 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 1/2.
Exercise 15.2
Q.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?
Sol.1) There are a total of 5 days. Shyam can go to the shop in 5 ways and Ekta can go to the shop in 5 ways.
Therefore, total number of outcomes = 5 × 5 = 25
(i) They can reach on the same day in 5 ways.
i.e., (t, t), (w, w), (th, th), (f, f), (s, s)
P (both will reach on same day) = 5/25 = 1/5
(ii) They can reach on consecutive days in these 8 ways - (t, w), (w, th), (th, f), (f, s), (w, t), (th, w), (f, th), (s, f).
Therefore, P (both will reach on consecutive days) = 8/25
(iii) P (both will reach on same day) 1/5 [(From (i)]
P (both will reach on different days) = 1 − (1/5) = 4/5
Q.2) A die is numbered in such a way that its faces show the number 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:
What is the probability that the total score is (i) even? (ii) 6? (iii) at least 6?
Total number of possible outcomes when two dice are thrown = 6 × 6 = 36
(i) Total times when the sum is even = 18
P (getting an even number) = 18/36 = 1/2
(ii) Total times when the sum is 6 = 4
P (getting sum as 6) = 4/36 = 1/9
(iii) Total times when the sum is at least 6 (i.e., greater than 5) = 15
P (getting sum at least 6) = 15/36 = 5/12
Q.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.
Sol.3) Let the number of blue balls be 𝑥.
Number of red balls = 5
Total number of balls = 𝑥 + 5
P (getting a red ball) = 5/𝑥+5
P (getting a blue ball) = 𝑥/𝑥+5
Given that,
2 (5/𝑥 + 5) = 𝑥/𝑥 + 5
10(𝑥 + 5) = 𝑥2 + 5𝑥
𝑥2 − 5𝑥 − 50 = 0
𝑥2 − 10𝑥 + 5𝑥 − 50 = 0
𝑥(𝑥 − 10) + 5(𝑥 − 10) = 0
(𝑥 − 10)(𝑥 + 5) = 0
Either 𝑥 − 10 = 0 or 𝑥 + 5 = 0
𝑥 = 10 or 𝑥 = −5
However, the number of balls cannot be negative.
Hence, number of blue balls = 10
Q.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 𝑥.
Sol.4) Total number of balls = 12
Total number of black balls = 𝑥
P (getting a black ball) = 𝑥/12
If 6 more black balls are put in the box, then
Total number of balls = 12 + 6 = 18
Total number of black balls = 𝑥 + 6
P (getting a black ball now) = 𝑥+6/18
According to the condition given in the question,
2 (𝑥/12) = 𝑥 + 6/18
3𝑥 = 𝑥 + 6
2𝑥 + 6
𝑥 = 3
Q.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 2/3. Find the number of blue balls in the jar.
Sol.5) Total number of marbles = 24
Let the total number of green marbles be 𝑥.
Then, total number of blue marbles = 24 − 𝑥
P (getting a given marble) = 𝑥/24
According to the condition given in the question,
𝑥/24 = 2/3
𝑥 = 16
Therefore, total number of green marbles in the jar = 16
Hence, total number of blue marbles = 24 − 𝑥 = 24 − 16 = 8
Either way the teacher or student will get the solution to the problem within 24 hours.