1
A group of 30 students in a class were asked about their favourite month of the year. The results are shown in the table below:
| Month | Number of Students |
|---|---|
| January | 3 |
| February | 2 |
| March | 4 |
| April | 2 |
| May | 3 |
| June | 2 |
| July | 1 |
| August | 2 |
| September | 3 |
| October | 2 |
| November | 3 |
| December | 3 |
| If a student is selected at random from this group, what is the probability that their favourite month is: | |
| (a) October? [1 mark] | |
| (b) A month with 31 days? [2 marks] | |
| (c) A month that falls in the second quarter of the year? [2 marks] |
2
Assertion (A): The probability of drawing a red ball from a bag containing 3 red and 2 blue balls is 3/5.Reason (R): The probability of an event is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes, assuming all outcomes are equally likely.(A) Both A and R are true and R is the correct explanation of A. [1 mark](B) Both A and R are true but R is not the correct explanation of A.(C) A is true but R is false.(D) A is false but R is true.
3
A survey was conducted in a locality about the preferred mode of transport for commuting to work. The data is as follows:
| Mode of Transport | Number of People |
|---|---|
| Car | 150 |
| Motorcycle | 200 |
| Bicycle | 75 |
| Public Bus | 250 |
| Walking | 125 |
| Based on this data, if a person is selected at random from this locality, what is the probability that the person prefers: | |
| (a) To commute by Motorcycle? [1 mark] | |
| (b) To commute by Car or Bicycle? [2 marks] | |
| (c) To commute by a mode that is NOT Walking? [2 marks] |
4
A game involves spinning an arrow on a circular board divided into 8 equal sectors, numbered 1 to 8. The arrow is equally likely to land on any sector.(i) What is the probability that the arrow lands on an even number? [1 mark](ii) What is the probability that the arrow lands on a number greater than 5? [1 mark](iii) What is the probability that the arrow lands on a prime number? [2 marks](iv) What is the probability that the arrow lands on a number that is a multiple of 3? 1mark
5
A bag contains 12 red balls and some blue balls. If the probability of drawing a blue ball at random is 2/5, find the number of blue balls in the bag. 3marks
6
A teacher prepared 50 flashcards for a quiz. 20 cards have mathematical terms, 15 have scientific terms, and the rest have historical terms. One card is picked at random.(i) What is the probability that the card picked has a historical term? [2 marks](ii) What is the probability that the card picked has a scientific term? [1 mark](iii) What is the probability that the card picked is NOT a mathematical term? 2marks
7
Assertion (A): When a die is rolled, the probability of getting a number greater than 6 is 0.Reason (R): An impossible event has a probability of 0.(A) Both A and R are true and R is the correct explanation of A. [1 mark](B) Both A and R are true but R is not the correct explanation of A.(C) A is true but R is false.(D) A is false but R is true.
8
A company manufactures light bulbs. In a batch of 500 bulbs, 20 are found to be defective. If a bulb is selected at random from this batch, what is the probability that:(i) The selected bulb is defective? [1 mark](ii) The selected bulb is not defective? [1 mark](iii) The selected bulb is either defective or not defective? 1mark
9
In a survey of 100 students, it was found that 60 students like Maths, 50 like Science, and 30 like both Maths and Science. If a student is chosen at random from the surveyed group, what is the probability that the student:(i) Likes only Maths? [2 marks](ii) Likes only Science? [2 marks](iii) Likes neither Maths nor Science? 1mark
10
A two-digit number is formed by the digits 2, 3, 5, 7 and 9 without repetition.(i) What is the total number of possible two-digit numbers that can be formed? [1 mark](ii) What is the probability that the number formed is prime? [2 marks](iii) What is the probability that the number formed is an even number? [1 mark](iv) What is the probability that the number formed is greater than 70? 1mark
11
A letter is chosen at random from the word "MATHEMATICS".(i) What is the probability that the letter chosen is a vowel? [2 marks](ii) What is the probability that the letter chosen is a consonant? [1 mark](iii) What is the probability that the letter chosen is 'M'? [1 mark](iv) What is the probability that the letter chosen is NOT 'T'? 1mark
12
A shuttlecock manufacturing company produces shuttlecocks in two colours: red and white. Out of 500 shuttlecocks produced daily, 350 are red and the rest are white. A shuttlecock is selected at random. What is the probability that the selected shuttlecock is:(i) Red? [1 mark](ii) White? [1 mark](iii) Neither red nor white? 1mark
13
Consider the experiment of rolling two dice simultaneously.(i) Write down the total number of possible outcomes. [1 mark](ii) What is the probability of getting a sum of 10? [2 marks](iii) What is the probability of getting a sum of at least 11? 2marks
14
Assertion (A): The probability of getting a head when a fair coin is tossed is 1/2.Reason (R): The probability of an event is the number of outcomes favourable to the event divided by the total number of equally likely outcomes.(A) Both A and R are true and R is the correct explanation of A. [1 mark](B) Both A and R are true but R is not the correct explanation of A.(C) A is true but R is false.(D) A is false but R is true.
15
A survey of 40 families was conducted to find the relationship between the number of children and the number of female children. The data is presented below:
| Number of Female Children | Number of Families |
|---|---|
| 0 | 5 |
| 1 | 15 |
| 2 | 18 |
| 3 | 2 |
| If a family is selected at random, what is the probability that the family has: | |
| (a) Exactly 2 female children? [1 mark] | |
| (b) At least one female child? [2 marks] | |
| (c) No female children? [1 mark] | |
| (d) More than 2 female children? [1 mark] |
16
A card is drawn at random from a well-shuffled deck of 52 cards. What is the probability of drawing:(i) A black queen? [1 mark](ii) A face card? [2 marks](iii) A red card? [1 mark](iv) A card that is neither a spade nor a king? 2marks
17
Two players, Ajay and Vijay, play a badminton match. The probability of Ajay winning the match is 0.72. What is the probability of Vijay winning the match? 1mark
18
A quiz competition has two rounds. The probability of a student qualifying for the second round is 0.75. If 1200 students participated in the first round, how many students are expected to qualify for the second round? 2marks
19
A bag contains 5 red balls, 7 white balls and 3 green balls. One ball is drawn at random from the bag. What is the probability that the ball drawn is:(i) White? [1 mark](ii) Not green? [2 marks](iii) Red or green? 2marks
20
Consider the given spinner with 5 equal sectors numbered 1, 2, 3, 4, 5.[Diagram of a spinner with 5 equal sectors labeled 1, 2, 3, 4, 5]If the spinner is spun once, find the probability of landing on:(i) A number divisible by 2. [1 mark](ii) A prime number. [2 marks](iii) A number greater than 3. [1 mark](iv) A number less than 6. 1mark
Answers
1
Total number of students = 3+2+4+2+3+2+1+2+3+2+3+3 = 30.(a) Number of students whose favourite month is October = 2. P(October) = 2/30 = 1/15.(b) Months with 31 days are January, March, May, July, August, October, December. Number of students = 3+4+3+1+2+2+3 = 18. P(month with 31 days) = 18/30 = 3/5.(c) The second quarter of the year consists of April, May, June. Number of students = 2+3+2 = 7. P(month in second quarter) = 7/30.
2
(A) Both A and R are true and R is the correct explanation of A. The probability of getting a red ball is the number of red balls (3) divided by the total number of balls (3+2=5), which is 3/5. This aligns with the definition of theoretical probability.
3
Total number of people = 150 + 200 + 75 + 250 + 125 = 800.(a) Number of people who prefer Motorcycle = 200. P(Motorcycle) = 200/800 = 1/4.(b) Number of people who prefer Car or Bicycle = 150 + 75 = 225. P(Car or Bicycle) = 225/800 = 9/32.(c) Number of people who do NOT prefer Walking = 800 - 125 = 675. P(not Walking) = 675/800 = 27/32.
4
Total possible outcomes = 8 (sectors 1 to 8).(i) Even numbers are 2, 4, 6, 8. Number of favourable outcomes = 4. P(even) = 4/8 = 1/2.(ii) Numbers greater than 5 are 6, 7, 8. Number of favourable outcomes = 3. P(>5) = 3/8.(iii) Prime numbers are 2, 3, 5, 7. Number of favourable outcomes = 4. P(prime) = 4/8 = 1/2.(iv) Multiples of 3 are 3, 6. Number of favourable outcomes = 2. P(multiple of 3) = 2/8 = 1/4.
5
Let B be the number of blue balls. Total balls = 12 (red) + B (blue).Probability of drawing a blue ball = B / (12 + B).Given P(Blue) = 2/5.So, B / (12 + B) = 2/55B = 2(12 + B)5B = 24 + 2B3B = 24B = 8.There are 8 blue balls.
6
Total number of flashcards = 50.Number of mathematical terms = 20.Number of scientific terms = 15.Number of historical terms = 50 - (20 + 15) = 50 - 35 = 15.(i) P(historical term) = 15/50 = 3/10.(ii) P(scientific term) = 15/50 = 3/10.(iii) Number of cards that are NOT mathematical terms = 50 - 20 = 30. P(not mathematical) = 30/50 = 3/5.
7
(A) Both A and R are true and R is the correct explanation of A. Rolling a die once results in outcomes {1, 2, 3, 4, 5, 6}. There is no outcome greater than 6, hence it is an impossible event. The probability of an impossible event is 0.
8
Total number of bulbs = 500. Number of defective bulbs = 20.(i) P(defective) = Number of defective bulbs / Total bulbs = 20/500 = 1/25.(ii) Number of non-defective bulbs = 500 - 20 = 480. P(not defective) = 480/500 = 24/25.(iii) The event of selecting a bulb that is either defective or not defective is a sure event, as every bulb falls into one of these categories. P(defective or not defective) = 1. Alternatively, P(defective) + P(not defective) = 1/25 + 24/25 = 25/25 = 1.
9
Total students = 100.Let M be the set of students who like Maths, S be the set of students who like Science.n(M) = 60, n(S) = 50, n(M ∩ S) = 30.(i) Number of students who like only Maths = n(M) - n(M ∩ S) = 60 - 30 = 30. P(only Maths) = 30/100 = 3/10.(ii) Number of students who like only Science = n(S) - n(M ∩ S) = 50 - 30 = 20. P(only Science) = 20/100 = 1/5.(iii) Number of students who like Maths or Science or both = n(M ∪ S) = n(M) + n(S) - n(M ∩ S) = 60 + 50 - 30 = 80.Number of students who like neither Maths nor Science = Total students - n(M ∪ S) = 100 - 80 = 20.P(neither Maths nor Science) = 20/100 = 1/5.
10
Digits are {2, 3, 5, 7, 9}.(i) Total number of possible two-digit numbers without repetition = 5 choices for the tens digit × 4 choices for the units digit = 20.(ii) The two-digit numbers are: 23, 25, 27, 29, 32, 35, 37, 39, 52, 53, 57, 59, 72, 73, 75, 79, 92, 93, 95, 97.Prime numbers from this list: 23, 29, 37, 53, 59, 73, 79, 97. (Note: 5 is a digit, but 25, 35, 55, 75, 95 are not prime. 93 is divisible by 3. 92, 32, 52, 72 are even.)Number of prime numbers = 8. P(prime) = 8/20 = 2/5.(iii) Even numbers formed: 32, 52, 72, 92. Number of even numbers = 4. P(even) = 4/20 = 1/5.(iv) Numbers greater than 70: 72, 73, 75, 79, 92, 93, 95, 97. Number of numbers > 70 = 8. P(>70) = 8/20 = 2/5.
11
Word: MATHEMATICS. Total letters = 11.Letters: M(2), A(2), T(2), H(1), E(1), I(1), C(1), S(1).(i) Vowels: A, A, E, I. Number of vowels = 4. P(vowel) = 4/11.(ii) Consonants: M, M, T, T, H, C, S. Number of consonants = 7. P(consonant) = 7/11.(iii) Number of 'M's = 2. P('M') = 2/11.(iv) Number of 'T's = 2. Number of letters that are NOT 'T' = 11 - 2 = 9. P(not 'T') = 9/11.
12
Total shuttlecocks = 500. Red = 350. White = 500 - 350 = 150.(i) P(red) = 350/500 = 7/10.(ii) P(white) = 150/500 = 3/10.(iii) P(neither red nor white) = 0, as all shuttlecocks are either red or white.
13
When rolling two dice, the total number of possible outcomes is 6 × 6 = 36.(i) Total possible outcomes = 36.(ii) Outcomes with a sum of 10: (4, 6), (5, 5), (6, 4). Number of outcomes = 3. P(sum=10) = 3/36 = 1/12.(iii) Outcomes with a sum of at least 11:Sum = 11: (5, 6), (6, 5) - 2 outcomes.Sum = 12: (6, 6) - 1 outcome.Total outcomes with sum at least 11 = 2 + 1 = 3. P(sum ≥ 11) = 3/36 = 1/12.
14
(A) Both A and R are true and R is the correct explanation of A. A fair coin has two equally likely outcomes: head and tail. The probability of getting a head is 1 (favorable outcome) divided by 2 (total outcomes).
15
Total families = 40.(a) Number of families with exactly 2 female children = 18. P(exactly 2) = 18/40 = 9/20.(b) Number of families with at least one female child = 15 + 18 + 2 = 35. P(at least 1) = 35/40 = 7/8.(c) Number of families with no female children = 5. P(no female children) = 5/40 = 1/8.(d) Number of families with more than 2 female children = 2. P(>2 female children) = 2/40 = 1/20.
16
Total cards = 52.(i) Black queens: Queen of Clubs, Queen of Diamonds. Number = 2. P(black queen) = 2/52 = 1/26.(ii) Face cards: Jack, Queen, King in each of the 4 suits. Number = 3 × 4 = 12. P(face card) = 12/52 = 3/13.(iii) Red cards: Hearts and Diamonds. Number = 26. P(red card) = 26/52 = 1/2.(iv) Cards that are NOT spades and NOT kings.Total cards = 52.Spades = 13. Kings = 4. King of Spades is counted in both.Number of spades or kings = 13 + 4 - 1 = 16.Number of cards that are neither spade nor king = 52 - 16 = 36.P(neither spade nor king) = 36/52 = 9/13.
17
Let A be the event Ajay wins, V be the event Vijay wins. These are complementary events.P(A) = 0.72.P(V) = 1 - P(A) = 1 - 0.72 = 0.28.
18
Number of students in the first round = 1200.Probability of qualifying for the second round = 0.75.Expected number of students to qualify = Total students × Probability of qualifying= 1200 × 0.75 = 1200 × (3/4) = 900.
19
Total balls = 5 (red) + 7 (white) + 3 (green) = 15.(i) P(white) = Number of white balls / Total balls = 7/15.(ii) Number of balls that are not green = 15 - 3 = 12. P(not green) = 12/15 = 4/5.(iii) P(red or green) = P(red) + P(green) = (5/15) + (3/15) = 8/15.
20
Total outcomes = 5 (sectors 1 to 5).(i) Numbers divisible by 2: {2, 4}. Number of favourable outcomes = 2. P(divisible by 2) = 2/5.(ii) Prime numbers: {2, 3, 5}. Number of favourable outcomes = 3. P(prime) = 3/5.(iii) Numbers greater than 3: {4, 5}. Number of favourable outcomes = 2. P(>3) = 2/5.(iv) Numbers less than 6: {1, 2, 3, 4, 5}. Number of favourable outcomes = 5. P(<6) = 5/5 = 1.
