Probability - Exercise Solutions

Exercise 14.1
Key Concepts

Exercise 14.1 Solutions

1. Complete the following statements:

(i) Probability of an event E + Probability of the event 'not E' = 1.

(ii) The probability of an event that cannot happen is 0. Such an event is called an impossible event.

(iii) The probability of an event that is certain to happen is 1. 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 1.

(v) The probability of an event is greater than or equal to 0 and less than or equal to 1.

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: Not equally likely. The outcome depends on the condition of the car, battery, fuel, etc.

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

Answer: Not equally likely. The outcome depends on the player's skill, distance from the basket, etc.

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

Answer: Equally likely. If the person is guessing, both outcomes (right or wrong) are equally likely.

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

Answer: Equally likely. In the absence of any external factors, both outcomes are equally likely.

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?

Tossing a coin is considered fair because:

  • The coin is unbiased (symmetrical)
  • Both outcomes (head or tail) are equally likely
  • It eliminates any bias or preference
  • The outcome is completely random
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: (B) -1.5 cannot be the probability of an event because probability values always lie between 0 and 1 (inclusive).

5. If P(E) = 0.05, what is the probability of 'not E'?

We know that P(E) + P(not E) = 1

Therefore, P(not E) = 1 - P(E) = 1 - 0.05 = 0.95

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: Since the bag contains only lemon flavoured candies, the probability of taking out an orange flavoured candy is 0.

(ii) a lemon flavoured candy?

Answer: Since all candies are lemon flavoured, the probability of taking out a lemon flavoured candy is 1.

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?

Let E be the event that 2 students have the same birthday.

Then, P(not E) = 0.992

We know that P(E) + P(not E) = 1

Therefore, P(E) = 1 - P(not E) = 1 - 0.992 = 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?

Total number of balls = 3 + 5 = 8

(i) P(red ball) = Number of red balls / Total number of balls = 3/8

(ii) P(not red) = 1 - P(red) = 1 - 3/8 = 5/8

Alternatively, P(not red) = Number of black balls / Total number of balls = 5/8

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?

Total number of marbles = 5 + 8 + 4 = 17

(i) P(red) = 5/17

(ii) P(white) = 8/17

(iii) P(not green) = 1 - P(green) = 1 - 4/17 = 13/17

Alternatively, P(not green) = (5 + 8)/17 = 13/17

10. A piggy bank contains hundred 50p 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 50p coin? (ii) will not be a ₹5 coin?

Total number of coins = 100 + 50 + 20 + 10 = 180

(i) P(50p coin) = 100/180 = 5/9

(ii) P(not ₹5 coin) = 1 - P(₹5 coin) = 1 - 10/180 = 1 - 1/18 = 17/18

Alternatively, P(not ₹5 coin) = (100 + 50 + 20)/180 = 170/180 = 17/18

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. What is the probability that the fish taken out is a male fish?

Total number of fish = 5 + 8 = 13

P(male fish) = 5/13

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, and these are equally likely outcomes. What is the probability that it will point at

Total possible outcomes = 8

(i) P(8) = 1/8

(ii) Odd numbers are 1, 3, 5, 7 → 4 outcomes

P(odd number) = 4/8 = 1/2

(iii) Numbers greater than 2 are 3, 4, 5, 6, 7, 8 → 6 outcomes

P(number > 2) = 6/8 = 3/4

(iv) Numbers less than 9 are 1, 2, 3, 4, 5, 6, 7, 8 → 8 outcomes

P(number < 9) = 8/8 = 1

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

Total possible outcomes = 6

(i) Prime numbers on a die are 2, 3, 5 → 3 outcomes

P(prime number) = 3/6 = 1/2

(ii) Numbers between 2 and 6 are 3, 4, 5 → 3 outcomes

P(number between 2 and 6) = 3/6 = 1/2

(iii) Odd numbers on a die are 1, 3, 5 → 3 outcomes

P(odd number) = 3/6 = 1/2

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

Total possible outcomes = 52

(i) There are 2 red kings (hearts and diamonds)

P(king of red colour) = 2/52 = 1/26

(ii) Face cards are jack, queen, king. There are 3 face cards in each suit and 4 suits.

Total face cards = 3 × 4 = 12

P(face card) = 12/52 = 3/13

(iii) Red face cards: In hearts and diamonds, there are 3 face cards each.

Total red face cards = 3 + 3 = 6

P(red face card) = 6/52 = 3/26

(iv) There is only 1 jack of hearts

P(jack of hearts) = 1/52

(v) There are 13 spades in a deck

P(spade) = 13/52 = 1/4

(vi) There is only 1 queen of diamonds

P(queen of diamonds) = 1/52

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.

Total cards = 5

(i) P(queen) = 1/5

(ii) If the queen is drawn and put aside, then:

Remaining cards = 4

(a) P(ace) = 1/4

(b) P(queen) = 0 (since the queen has already been drawn and put aside)

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.

Total pens = 132 + 12 = 144

P(good pen) = 132/144 = 11/12

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?

(i) Total bulbs = 20

Defective bulbs = 4

P(defective) = 4/20 = 1/5

(ii) If the bulb drawn in (i) is not defective and is not replaced:

Remaining bulbs = 20 - 1 = 19

Remaining non-defective bulbs = 16 - 1 = 15

P(not defective) = 15/19

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.

Total discs = 90

(i) Two-digit numbers from 1 to 90: 10 to 90 → 81 numbers

P(two-digit number) = 81/90 = 9/10

(ii) Perfect squares from 1 to 90: 1, 4, 9, 16, 25, 36, 49, 64, 81 → 9 numbers

P(perfect square) = 9/90 = 1/10

(iii) Numbers divisible by 5 from 1 to 90: 5, 10, 15, ..., 90 → 18 numbers

P(divisible by 5) = 18/90 = 1/5

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?

Total possible outcomes = 6

(i) Number of faces with A = 2

P(A) = 2/6 = 1/3

(ii) Number of faces with D = 1

P(D) = 1/6

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 1m?

Area of rectangle = 3 × 2 = 6 m²

Radius of circle = 1/2 = 0.5 m

Area of circle = πr² = π(0.5)² = 0.25π m²

P(landing in circle) = Area of circle / Area of rectangle = 0.25π / 6 = π/24

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

Total pens = 144

Defective pens = 20

Good pens = 144 - 20 = 124

(i) P(she will buy) = P(good pen) = 124/144 = 31/36

(ii) P(she will not buy) = P(defective pen) = 20/144 = 5/36

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

From Example 13, when two dice are thrown, the possible sums are from 2 to 12.

Event: 'Sum on 2 dice' 2 3 4 5 6 7 8 9 10 11 12
Probability 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36

(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? Justify your answer.

Answer: No, I do not agree with this argument. Although there are 11 possible sums (2 to 12), these outcomes are not equally likely. For example, the sum 2 can occur in only 1 way (1,1), while the sum 7 can occur in 6 ways (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). Therefore, the probability of each sum is not the same.

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.

When a coin is tossed 3 times, the possible outcomes are:

HHH, HHT, HTH, THH, HTT, THT, TTH, TTT → 8 equally likely outcomes

Hanif wins if: HHH or TTT → 2 outcomes

P(Hanif wins) = 2/8 = 1/4

P(Hanif loses) = 1 - P(Hanif wins) = 1 - 1/4 = 3/4

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

When a die is thrown twice, total possible outcomes = 6 × 6 = 36

(i) P(5 will not come up either time) = 1 - P(5 comes up at least once)

Let's find P(5 comes up at least once):

Outcomes with no 5: When both dice show numbers from {1,2,3,4,6} → 5 × 5 = 25 outcomes

P(no 5) = 25/36

Therefore, P(5 will not come up either time) = 25/36

(ii) P(5 will come up at least once) = 1 - P(no 5) = 1 - 25/36 = 11/36

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.

Answer: Not correct. When two coins are tossed, the possible outcomes are: HH, HT, TH, TT. These are 4 equally likely outcomes, not 3. The probability of two heads is 1/4, two tails is 1/4, and one of each is 2/4 = 1/2.

(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 1/2.

Answer: Correct. When a die is thrown, the outcomes 1, 2, 3, 4, 5, 6 are equally likely. Odd numbers are 1, 3, 5 (3 outcomes) and even numbers are 2, 4, 6 (3 outcomes). So P(odd) = 3/6 = 1/2.

Key Concepts in Probability

Probability Formula

P(E) = Number of outcomes favourable to E / Number of all possible outcomes

This formula applies when all outcomes are equally likely.

Important Properties

  • 0 ≤ P(E) ≤ 1 for any event E
  • P(sure event) = 1
  • P(impossible event) = 0
  • P(E) + P(not E) = 1
  • The sum of probabilities of all elementary events of an experiment is 1

Types of Events

  • Elementary Event: An event having only one outcome
  • Sure Event: An event that is certain to happen (probability = 1)
  • Impossible Event: An event that cannot happen (probability = 0)
  • Complementary Events: For any event E, the event 'not E' is called the complement of E, denoted by \(\overline{E}\)

Important Points

  • Probability is a theoretical concept that predicts the likelihood of events
  • Empirical probability is based on actual experiments and observations
  • As the number of trials increases, empirical probability tends to approach theoretical probability
  • Probability finds applications in various fields like statistics, genetics, economics, and more

Note: The probability values always range between 0 and 1, inclusive. A probability of 0 means the event is impossible, while a probability of 1 means the event is certain to occur.