The Mathematics of Maybe: Introduction to Probability — Class 9 Mathematics

What is the probability of an impossible event?

It is , the lowest value on the probability scale.

What is the probability of an event that is certain?

It is , the highest value on the probability scale.

Write the sample space for rolling a standard die.

It is , with sample size .

What is the theoretical probability of rolling a on a fair die?

It is , one favourable outcome out of six.

Define experimental probability.

It is the number of times an event occurs divided by the total number of trials performed.

Find when a letter is picked from the word PROBABILITY.

There are B's out of letters, so .

What is the probability of getting an even number on a die?

The even outcomes are , so .

What does the Gambler's Fallacy mistakenly assume?

That a run of one outcome makes the opposite outcome more likely; in fact each trial is independent.

What is the sample space when two coins are tossed?

It is , with four equally likely outcomes.

What happens to experimental probability as the number of trials grows large?

By the Law of Large Numbers it gets closer and closer to the theoretical probability.

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.

Ranking: - Certain. Reason: The calendar cycle is fixed; Sunday is always followed by Monday, so the event must happen.

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.

Ranking: - Impossible (in practice). Reason: Mumbai has a tropical hot climate and is at sea level; snowfall has never been recorded there, and July is monsoon season with high temperatures, so snow is virtually impossible.

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.

Ranking: - Impossible or extremely less likely, effectively . Reason: Elephants do not normally enter school classrooms; unless your school is next to a forest or circus, the chance is negligible.

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.

Ranking: More likely, around to but not certain. Reason: If you attend school and have friends there, it is very probable you will meet and greet someone, but absence, holiday or illness could prevent it, so it is not .

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.

From the sample, number of green sweets is out of total . Experimental probability or about .

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.

Probability of yellow from sample is . Estimated number in is yellow sweets.

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?

Number preferring Arts Club is out of . So or .

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.

. For students, estimated number is students.

3. Toss a coin 20 times and record the result each time (heads or tails). (i) How many times did you get heads?

This is an experimental activity. Answers will vary. As an example, a student might get 11 heads out of 20 tosses. You should count and write your actual observed number.

3. Toss a coin 20 times and record the result each time (heads or tails). (ii) How many times did you get tails?

This will be minus the number of heads. For example, if you got 11 heads, you got 9 tails. Record your actual count.

3. Toss a coin 20 times and record the result each time (heads or tails). (iii) Calculate the experimental probability of getting heads.

Experimental probability . For example, with 11 heads, . Your value depends on your experiment.

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?

Theoretical probability for a fair coin is still regardless of previous results. The coin has no memory.

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.

This is an experimental task. After 100 tosses, count where . Then , , . Typically is largest because the side area is largest. Example: if results are 15 bottom, 15 top, 70 side, then probabilities are . Your numbers will vary.

5. What is the probability of getting an even number when rolling a fair 6-sided die?

Sample space is . Favourable outcomes for even are which is 3 outcomes. So or .

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’?

Experimental probability or .

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’?

For a fair die, each face is equally likely. or about .

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?

They differ because experimental probability is based on a small number of trials (12) which can have random variation. Theoretical probability is the ideal long-run value. By the Law of Large Numbers, as the number of trials increases to 60, 600 or 6000, the experimental probability will tend to get closer to .

1. When a single 6-sided die is rolled, what is the total number of possible outcomes in the sample space?

The sample space is . The number of elements is .

2. For the following experiments write down the sample space S. (i) Rolling a die and tossing a coin together.

Die gives and coin gives . Combined sample space: , so .

2. For the following experiments write down the sample space S. (ii) Choosing a random integer between – 5 and + 5.

If 'between' is exclusive of -5 and +5, with . If inclusive, with . For class 9, inclusive interpretation is often intended as .

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.

If we list by colour only, with . If we distinguish each ball, with .

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.

Using S=Samosa, P=Pakora, B=Bhaji and C=Chai, L=Lassi: , so .

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.’

Event i.e., Samosa with Chai and Samosa with Lassi. It has 2 outcomes.

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.

Level 1 branches from Basket A: Apple (A), Orange1 (O1), Orange2 (O2) or simply A with probability and O with probability . From each branch, level 2 branches to Banana (B) and Mango (M) each with probability . Paths are A-B, A-M, O-B, O-M.

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.

If oranges are identical, with 4 distinct pairs. Counting individual oranges, there are 6 equally likely pairs: (A,B),(A,M),(O1,B),(O1,M),(O2,B),(O2,M).

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?

Favourable is (Apple, Banana) only. With 4 distinct combinations if oranges considered same type, . With individual fruits equally likely, . Most textbooks give assuming each fruit equally likely.

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?

First pick colours: R, B, G. Second pick also R, B, G. Tree: first level 3 branches R (3/9), B (4/9), G (2/9). From each, second level again 3 branches R, B, G. There are combined ordered outcomes: (R,R),(R,B),(R,G),(B,R),(B,B),(B,G),(G,R),(G,B),(G,G).

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?

With replacement, picks are independent. , , . So or about .

1. Fill in the blanks. (i) The probability of an impossible event is _______.

1. Fill in the blanks. (ii) The set of all possible outcomes of a random experiment is called the __________.

sample space (denoted )

1. Fill in the blanks. (iii) The probability of an event that is certain to happen is _______.

1. Fill in the blanks. (iv) Tossing a fair coin has a probability of ______ for getting heads.

or .

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).

The count 15 is the frequency. The relative frequency is or . So blank is relative frequency, value is .

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.

Not equally likely. The car is designed to start, so is much higher than , except if faulty.

3. Which of the following experiments have equally likely outcomes? Explain. (ii) Tossing a fair coin once.

Equally likely. For a fair, unbiased coin, .

3. Which of the following experiments have equally likely outcomes? Explain. (iii) Rolling a fair 6-sided die.

Equally likely. Each face to has probability .

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.

Not equally likely by colour. and . If considering each individual marble, they are equally likely.

3. Which of the following experiments have equally likely outcomes? Explain. (v) A baby is born. It is a boy or a girl.

Approximately equally likely in theory, . In real population data it is slightly biased (~ boy), but for school problems treated as equally likely.

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?

Sample space , . Event , . So .

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?

, . Even numbers , . .

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?

. Favourable , . .

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?

Total balls . Non-red balls are blue + green = . So . Alternatively .

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?

, . Exactly two heads: , . .

5. A bag has 3 candies: strawberry, lemon, and mint. One is picked at random. What is the probability of picking a strawberry candy?

, . .

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.

Possible outfits : Table rows shirts, columns pants: (Red, Jeans), (Red, Khakis), (Red, Shorts), (Blue, Jeans), (Blue, Khakis), (Blue, Shorts). In table form: | | Jeans | Khakis | Shorts | | Red | R-J | R-K | R-S | | Blue | B-J | B-K | B-S |

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.

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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.

Cases between 4001-9000 plus 9001-14000 = . So . If inclusive of 4000-14000, same value.

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.

.

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?

Word PEACE has 5 letters: P,E,A,C,E. Favourable P,C,E. Count = P(1)+C(1)+E(2)=4 cards. So .

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?

Not E = P,A,C = 3 cards. . Alternatively .

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?

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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?

Odd numbers =4 out of 8. .

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?

Numbers greater than 2: =6. .

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?

All numbers 1-8 are less than 9. So certain.

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?

Multiples of 3 in range: =2. .

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?

Without replacement. Tree: First level R , B . Second level: If R first, then R , B ; if B first, then R , B . So .

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?

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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.

Probability 0 (impossible): Example - sum is 13 or sum is 1, or both dice show 7. Since max sum with two 6-sided dice is 12 and min is 2, . Probability 1 (certain): Example - sum is between 2 and 12 inclusive, or sum is less than 13. So .

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?

Sample space ordered pairs. Prime numbers >5 up to 12 are . Count pairs: sum 7 =6 pairs (1,6),(2,5),(3,4),(4,3),(5,2),(6,1); sum 11 =2 pairs (5,6),(6,5). Total favourable 8. So . If including prime 5 is excluded by condition, otherwise would add 4 more for sum 5.

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?

Total 9 balls. Total ways or sequential. . So . Equivalently .

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?

Sample space 8 outcomes. Condition: first = H and exactly two H total => outcomes \{HHT,HTH\} =2. So .

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?

Total permutations . Even numbers end with 2 or 4. For each even ending, remaining 3 digits permute in ways. So favourable . .

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?

For each question , . Exactly 2 correct out of 3: choose which 2 are correct ways. So .

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.

With replacement: first level 1,2,3,4 each . Second level again 1,2,3,4. Sample space = , size 16.

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.

Without replacement: first ball 4 options, second ball 3 remaining. Sample space = , excludes doubles like (1,1). Size 12.

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?

With replacement size is . Without replacement size is .

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.

Coin , card . Combined sample space , .

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}

Not a valid sample space. It omits 0 heads (TTT) which is possible, and X=number of heads ranges 0-3. So incomplete.

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}

Not valid. Omits 3 heads (HHH) which is possible, so it does not include every possible outcome.

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}

Not valid. Includes 4 which is impossible when tossing only 3 coins; sample space must not contain impossible outcomes.

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}

Valid sample space. It lists exactly all possible numbers of heads 0 to 3, each once, with no missing or extra outcomes. So (iv) is correct.

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?

Geometric probability = . Rectangle is . Circle diameter so radius , area . So or about .