Class 9 › Mathematics › Ganita Manjari The Mathematics of Maybe: Introduction to Probability Class 9 Mathematics · Ganita Manjari · Chapter 7
Summary Questions & answers Practise this chapter → This chapter introduces probability as a measure of how likely a random event is to occur, on a scale from \(0\) (impossible) to \(1\) (certain), with \(0 \le P(E) \le 1\). Randomness means the outcome of an experiment such as tossing a coin or rolling a die cannot be predicted, though all possible outcomes are known. There are two objective ways to estimate probability. Experimental probability uses data: \(\dfrac{\text{number of times the event occurred}}{\text{total number of trials}}\), also called relative frequency. Theoretical probability assumes equally likely outcomes: \(\dfrac{\text{number of favourable outcomes}}{\text{number of possible outcomes}}\). By the Law of Large Numbers, experimental probability tends toward the theoretical value as trials increase. The chapter warns against the Gambler's Fallacy, since each trial is independent and has no memory. The sample space \(S\) is the set of all outcomes, its size \(n(S)\), and an event is a subset of \(S\). Tree diagrams list and find probabilities of outcomes in multi-step experiments. Statistical sampling estimates population proportions from a representative sample.
Probability
A number from \(0\) to \(1\) measuring the likelihood that an event occurs.
Sample space
The set \(S\) of all possible outcomes of a random experiment.
Event
Any subset of the sample space — one or more outcomes of interest.
Experimental probability
An estimate equal to \(\dfrac{\text{times event occurred}}{\text{total trials}}\), based on observed data.
Theoretical probability
For equally likely outcomes, \(\dfrac{\text{favourable outcomes}}{\text{total outcomes}}\).
Gambler's Fallacy
The mistaken belief that past results change the probability of an independent future trial.
Q1 What is the probability of an impossible event? Answer Q2 What is the probability of an event that is certain? Answer Q3 Write the sample space for rolling a standard die. Answer Q4 What is the theoretical probability of rolling a \(4\) on a fair die? Answer Q5 Define experimental probability. Answer Q6 Find \(P(B)\) when a letter is picked from the word PROBABILITY. Answer Q7 What is the probability of getting an even number on a die? Answer Q8 What does the Gambler's Fallacy mistakenly assume? Answer Q9 What is the sample space when two coins are tossed? Answer Q10 What happens to experimental probability as the number of trials grows large? Answer Q1 1. Rank the following events on a scale from 0 (Impossible) to 1 (Certain). Label each event: Impossible, less likely, equally likely (even chance), more likely, certain. Give reasons why you gave each event its ranking. (i) The next Monday will come after Sunday. Answer Q2 1. Rank the following events on a scale from 0 (Impossible) to 1 (Certain). Label each event: Impossible, less likely, equally likely (even chance), more likely, certain. Give reasons why you gave each event its ranking. (ii) It will snow in Mumbai in July. Answer Q3 1. Rank the following events on a scale from 0 (Impossible) to 1 (Certain). Label each event: Impossible, less likely, equally likely (even chance), more likely, certain. Give reasons why you gave each event its ranking. (iii) An elephant will walk through your classroom today. Answer Q4 1. Rank the following events on a scale from 0 (Impossible) to 1 (Certain). Label each event: Impossible, less likely, equally likely (even chance), more likely, certain. Give reasons why you gave each event its ranking. (iv) You will greet at least one friend at school tomorrow. Answer Q1 1. A teacher mixes a large bag of sweets of different colours and randomly selects a sample of 30 sweets. She counts the number of sweets of each colour: 10 red sweets | 8 green sweets | 7 yellow sweets | 5 blue sweets (i) Calculate the probability that a randomly picked sweet from the sample is green. Answer Q2 1. A teacher mixes a large bag of sweets of different colours and randomly selects a sample of 30 sweets. She counts the number of sweets of each colour: 10 red sweets | 8 green sweets | 7 yellow sweets | 5 blue sweets (ii) If there are 600 sweets in total in the large bag, estimate how many are likely to be yellow, based on the sample results. Answer Q3 2. A survey is conducted at a school where a random sample of 40 students is asked about their favourite club. The responses are: 14 students: Science Club | 11 students: Arts Club | 9 students: Sports Club | 6 students: Debate Club Assume there are 800 students in the whole school. (i) What is the probability that a randomly chosen student from the sample prefers the Arts Club? Answer Q4 2. A survey is conducted at a school where a random sample of 40 students is asked about their favourite club. The responses are: 14 students: Science Club | 11 students: Arts Club | 9 students: Sports Club | 6 students: Debate Club Assume there are 800 students in the whole school. (ii) Using the sample results, estimate how many students in the whole school are likely to prefer the Sports Club. Answer Q5 3. Toss a coin 20 times and record the result each time (heads or tails). (i) How many times did you get heads? Answer Q6 3. Toss a coin 20 times and record the result each time (heads or tails). (ii) How many times did you get tails? Answer Q7 3. Toss a coin 20 times and record the result each time (heads or tails). (iii) Calculate the experimental probability of getting heads. Answer Q8 3. Toss a coin 20 times and record the result each time (heads or tails). (iv) If you toss the coin once more, what is the probability of getting tails? Answer Q9 4. Toss a paper cup into the air 100 times. After each toss record whether the cup lands on its bottom, upside down on its top or on its side (See Fig. 7.5). Assign probabilities to the outcomes by using experimental probability. Answer Q10 5. What is the probability of getting an even number when rolling a fair 6-sided die? Answer Q11 6. Suppose you roll a 6-sided die 12 times and get a ‘3’ three times. (i) What is the experimental probability of rolling a ‘3’? Answer Q12 6. Suppose you roll a 6-sided die 12 times and get a ‘3’ three times. (ii) What is the theoretical probability of rolling a ‘3’? Answer Q13 6. Suppose you roll a 6-sided die 12 times and get a ‘3’ three times. (iii) Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times? Answer Q1 1. When a single 6-sided die is rolled, what is the total number of possible outcomes in the sample space? Answer Q2 2. For the following experiments write down the sample space S. (i) Rolling a die and tossing a coin together. Answer Q3 2. For the following experiments write down the sample space S. (ii) Choosing a random integer between – 5 and + 5. Answer Q4 2. For the following experiments write down the sample space S. (iii) A box containing 5 green and 7 red balls. One ball is drawn at random. Answer Q5 3. In a village fair, there are 3 popular snacks available: Samosa, Pakora, and Bhaji. For drinks, villagers can choose either Chai or Lassi. (i) List the sample space of all possible snack and drink combinations a person could choose at the fair. Answer Q6 3. In a village fair, there are 3 popular snacks available: Samosa, Pakora, and Bhaji. For drinks, villagers can choose either Chai or Lassi. (ii) List the event ‘Selecting Samosa as a snack.’ Answer Q1 1. There are two fruit baskets A and B. Basket A has one apple and two oranges. Basket B has one banana and one mango. You randomly pick one fruit from each basket. (i) Draw a tree diagram showing all possible pairs of fruits. Answer Q2 1. There are two fruit baskets A and B. Basket A has one apple and two oranges. Basket B has one banana and one mango. You randomly pick one fruit from each basket. (ii) List the sample space. Answer Q3 1. There are two fruit baskets A and B. Basket A has one apple and two oranges. Basket B has one banana and one mango. You randomly pick one fruit from each basket. (iii) What is the probability of picking one apple and one banana? Answer Q4 2. Let us say that you have a box containing 3 red pens, 4 black pens and 2 green pens. You pick a pen (without looking) from the box and put it back. Then your friend does the same. (i) What are the possible outcomes of the pen colours? Can you draw a tree diagram representing the possible outcomes? Answer Q5 2. Let us say that you have a box containing 3 red pens, 4 black pens and 2 green pens. You pick a pen (without looking) from the box and put it back. Then your friend does the same. (ii) Can you use the tree diagram to guess the probability that both you and your friend pick pens of the same colour? Answer Q1 1. Fill in the blanks. (i) The probability of an impossible event is _______. Answer Q2 1. Fill in the blanks. (ii) The set of all possible outcomes of a random experiment is called the __________. Answer Q3 1. Fill in the blanks. (iii) The probability of an event that is certain to happen is _______. Answer Q4 1. Fill in the blanks. (iv) Tossing a fair coin has a probability of ______ for getting heads. Answer Q5 2. In a survey of 50 students, 15 students said they liked football. The number of students who like football is 15, and the ________ (frequency/relative frequency) is __________ (fill in the fraction or decimal). Answer Q6 3. 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 Q7 3. Which of the following experiments have equally likely outcomes? Explain. (ii) Tossing a fair coin once. Answer Q8 3. Which of the following experiments have equally likely outcomes? Explain. (iii) Rolling a fair 6-sided die. Answer Q9 3. Which of the following experiments have equally likely outcomes? Explain. (iv) Choosing a marble randomly from a bag that contains 3 red marbles and 7 blue marbles. Answer Q10 3. Which of the following experiments have equally likely outcomes? Explain. (v) A baby is born. It is a boy or a girl. Answer Q11 4. Write the sample space and calculate the probability based on the given information. (i) Two coins are tossed at the same time. What is the probability of getting at least one head? Answer Q12 4. Write the sample space and calculate the probability based on the given information. (ii) Ten identical cards numbered 1 to 10 are placed in a box. One card is drawn at random. What is the probability of drawing a card with an even number? Answer Q13 4. Write the sample space and calculate the probability based on the given information. (iii) A die is rolled once. What is the probability of getting a number greater than 4? Answer Q14 4. Write the sample space and calculate the probability based on the given information. (iv) A bag contains 3 red balls, 2 blue balls, and 1 green ball. One ball is picked at random. What is the probability that it is not red? Answer Q15 4. Write the sample space and calculate the probability based on the given information. (v) Three coins are tossed simultaneously. What is the probability of getting exactly two heads? Answer Q16 5. A bag has 3 candies: strawberry, lemon, and mint. One is picked at random. What is the probability of picking a strawberry candy? Answer Q17 6. A child has 2 shirts (one red and one blue) and 3 types of pants (jeans, khakis, and shorts). List all the possible combinations of outfits consisting of one shirt and one pair of pants. Display your answer in a table format. Answer Q18 7. A tyre company records distances before replacement in 1000 cases. Distance (km) Less than 4000 | 4001 to 9000 | 9001 to 14000 | More than 14000 Number of cases 20 | 210 | 325 | 445 Find the probability that a randomly chosen tyre lasts: (i) Less than 4000 km. Answer Q19 7. A tyre company records distances before replacement in 1000 cases. Find the probability that a randomly chosen tyre lasts: (ii) Between 4000 and 14000 km. Answer Q20 7. A tyre company records distances before replacement in 1000 cases. Find the probability that a randomly chosen tyre lasts: (iii) More than 14000 km. Answer Q21 8. The letters of the word ‘PEACE’ are placed on cards. Leela draws a card without looking. (i) What is the probability that it is a P, E or C? Answer Q22 8. The letters of the word ‘PEACE’ are placed on cards. Leela draws a card without looking. (ii) What is the probability that it is not an E? Answer Q23 9. A game of chance consists of spinning an arrow (see Fig. 7.7.) 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 (i) 8? Answer Q24 9. A game of chance consists of spinning an arrow (see Fig. 7.7.) 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 (ii) An odd number? Answer Q25 9. A game of chance consists of spinning an arrow (see Fig. 7.7.) 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 (iii) A number greater than 2? Answer Q26 9. A game of chance consists of spinning an arrow (see Fig. 7.7.) 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 (iv) A number less than 9? Answer Q27 9. A game of chance consists of spinning an arrow (see Fig. 7.7.) 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 (v) A multiple of 3? Answer Q28 10. A basket contains 4 red balls and 5 blue balls. One ball is drawn and laid aside, and a second ball is drawn. Draw a tree diagram to represent the possible outcomes and probabilities. Use the tree diagram to answer the following questions. (i) What is the probability of drawing a red ball and then a blue ball? Answer Q29 10. A basket contains 4 red balls and 5 blue balls. One ball is drawn and laid aside, and a second ball is drawn. Draw a tree diagram to represent the possible outcomes and probabilities. Use the tree diagram to answer the following questions. (ii) What is the probability of drawing 2 blue balls? Answer Q30 11. I throw a pair of 6-sided dice. Write down an event that has a probability of 0 and an outcome that has a probability of 1. Answer Q31 12. Write the sample space and calculate the probability based on the given information. (i) Two dice are rolled. What is the probability that the sum is a prime number greater than 5? Answer Q32 12. Write the sample space and calculate the probability based on the given information. (ii) A bag contains 4 red, 3 green, and 2 blue balls. Two balls are drawn without replacement. What is the probability that both are of different colours? Answer Q33 12. Write the sample space and calculate the probability based on the given information. (iii) Three coins are tossed. What is the probability that the first coin shows heads and exactly two heads occur in total? Answer Q34 12. Write the sample space and calculate the probability based on the given information. (iv) A four-digit number is formed using the digits 1, 2, 3, and 4 with no repetition. What is the probability that the number is even? Answer Q35 12. Write the sample space and calculate the probability based on the given information. (v) A student takes a multiple-choice test with 3 questions, each having 4 options (A, B, C, D), with only one correct answer. What is the probability that the student guesses and gets exactly 2 answers correct? Answer Q36 13. A box contains 4 balls numbered 1 to 4. Record a sample space using a tree diagram for the following experiments: (i) A ball is drawn, and the number is recorded. Then the ball is returned, and a second ball is drawn and recorded. Answer Q37 13. A box contains 4 balls numbered 1 to 4. Record a sample space using a tree diagram for the following experiments: (ii) A ball is drawn and recorded. Without replacing the first ball, the experimenter draws and records a second ball. Answer Q38 13. A box contains 4 balls numbered 1 to 4. Record a sample space using a tree diagram for the following experiments: (iii) What are the sizes of these two sample spaces? Answer Q39 14. List the elements of a sample space for the simultaneous tossing of a coin and drawing of a card from a set of 6 cards numbered 1 through 6. Answer Q40 15. Three coins are tossed, and the number of heads is recorded. Which of the following lists is a sample space for this experiment? Why do the other lists fail to qualify as a sample space? (i) {1, 2, 3} Answer Q41 15. Three coins are tossed, and the number of heads is recorded. Which of the following lists is a sample space for this experiment? Why do the other lists fail to qualify as a sample space? (ii) {0, 1, 2} Answer Q42 15. Three coins are tossed, and the number of heads is recorded. Which of the following lists is a sample space for this experiment? Why do the other lists fail to qualify as a sample space? (iii) {0, 1, 2, 3, 4} Answer Q43 15. Three coins are tossed, and the number of heads is recorded. Which of the following lists is a sample space for this experiment? Why do the other lists fail to qualify as a sample space? (iv) {0, 1, 2, 3} Answer Q44 16. Suppose you drop a dye at random on the rectangular region shown in Fig. 7.8. What is the probability that it will land inside the circle with a diameter of 1 m? Answer
The probability of an impossible event is:
A \(0\) B \(0.5\) C \(-1\) D \(1\)
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