The Mathematics of Maybe: Introduction to Probability Class 9 Notes and Solutions

The Mathematics of Maybe: Introduction to Probability

7.1 What is Probability?

Probability is a special kind of measurement. While a ruler measures length and a weighing scale measures mass, probability measures how likely an event is to happen. It helps us turn uncertainty into a number we can understand.

Think about these questions:

• Will it rain this evening?
• Will our school team win the hockey match?
• Who will be picked in the lucky draw?

These are random events. We know what could happen, but we cannot be 100 % sure what will happen. When you say, "It is unlikely to rain because the sun is shining," you are giving a personal or subjective probability based on what you see. In this chapter, we learn how to measure chance more objectively.

7.1.1 What is Randomness?

Randomness means an action or situation where the exact result cannot be predicted, even though we know all the possible results.

• Tossing a coin: The result is either Heads or Tails, but you cannot know which one will appear on a single toss.
• Rolling a die: The top face can show 1, 2, 3, 4, 5, or 6. You cannot say which number will appear before you roll.

An experiment that can be repeated and gives unpredictable results each time is called a random experiment or trial. In a lucky draw, every student has an equal chance of being chosen, but no one knows who will be picked in advance.

7.1.2 The Probability Scale

Probability is measured on a scale from 0 to 1.

• 0 means the event is impossible.
• 1 means the event is certain.
• 0.5 means there is an even chance — the event is just as likely to happen as not.

If your school has a 0.75 probability of winning a match, it means there is a 75 % chance of victory. Most real-life probabilities fall somewhere between 0 and 1.

7.2 Measuring Probability Objectively

Instead of guessing, we can use two objective methods:

1. Experimental probability: Collect data by performing an experiment many times or looking at past records. We calculate the relative frequency of the event.
2. Theoretical probability: Use reasoning. We assume every outcome is equally likely in a fair situation and count favorable outcomes.

7.2.1 Experimental Probability

In an experiment, each result is called an outcome. The list of every possible outcome is called the sample space.

• Coin toss sample space: S = {H, T}
• Die roll sample space: S = {1, 2, 3, 4, 5, 6}

If you roll a die 50 times and a 4 appears 8 times, the experimental probability is 850 = 0.16. This value is also called the relative frequency.

7.2.2 Theoretical Probability

Theoretical probability tells us what we expect in a perfectly fair world. It does not need an experiment.

For a fair six-sided die:

• P(rolling a 4) = 16 ≈ 0.167
• P(rolling an even number) = 36 = 12

7.2.3 Analysing Statistical Data Using Probability

Businesses and scientists use collected data to estimate probabilities. If a class survey shows 20 out of 50 students prefer mango, the probability that a randomly chosen student from that class likes mango is:

To guess the needs of a whole school of 1,500 students, we apply the same rate: about 40 % of 1,500 = 600 mangoes. Using a larger and more representative sample makes the estimate more trustworthy. This process is called sampling.

Gambler's Fallacy

Some people think that if a coin lands on heads many times in a row, tails becomes "due." This is a mistake. A coin has no memory. Every flip is a fresh start. The probability of tails on the next flip is still 12.

Believing that past random events change the odds of future independent events is called the Gambler's Fallacy. In Snakes and Ladders, each die roll is independent. Three sixes in a row do not make a fourth six less likely.

Fair and Unbiased

A coin is called fair or unbiased when it is perfectly balanced, so neither side is favored. A random toss means the coin is thrown freely without cheating or interference.

7.3 Elements of Probability

7.3.1 Sample Space

The sample space, written as S, is the complete set of all possible outcomes of a random experiment. The number of elements inside it is called the sample size, written as n(S).

Rules for writing a sample space:

• Every possible outcome must be listed.
• Do not repeat the same outcome.
• n(S) is the total count of unique outcomes.

If a question needs more detail, expand the sample space. For example, instead of just {Rain, No Rain}, you could use {No Rain, Drizzle, Light Rain, Heavy Rain} to match the level of detail required.

7.3.2 Events

An event is any single result or group of results we are interested in. It is a smaller set picked from the sample space.

• Two coins: Event "at least one head" = {HH, HT, TH}
• One die: Event "number greater than 4" = {5, 6}
• Fruit basket {Apple, Banana, Orange}: Event "yellow fruit" = {Banana}

7.4 Tree Diagrams

A tree diagram is a picture that helps us list every possible result of a multi-step experiment. Each branch shows one possible outcome, and new branches grow from it for the next step.

Every complete path from the start to the tip of a branch is one full outcome. Tree diagrams also let us write probabilities on each branch.

For the experiment above, the sample space is S = {HH, HT, TH, TT}.

The probability of getting two heads (HH) is:

End-of-Chapter Exercises

1. Fill in the blanks.

(i) The probability of an impossible event is 0.<
(ii) The set of all possible outcomes of a random experiment is called the sample space.<
(iii) The probability of an event that is certain to happen is 1.<
(iv) Tossing a fair coin has a probability of 12 for getting heads.

2. In a survey of 50 students, 15 said they liked football. The number of students who like football is 15, and the relative frequency is 1550 = 0.3.

3. Which experiments have equally likely outcomes? Explain.

(i) A driver starts a car — No. Mechanical condition and battery life can make one outcome more likely.
(ii) Tossing a fair coin — Yes. The coin is balanced, so heads and tails have the same chance.
(iii) Rolling a fair 6-sided die — Yes. Each face has the same shape and weight.
(iv) Choosing a marble from a bag with 3 red and 7 blue — No. There are more blue marbles, so blue is more likely.
(v) A baby is born, boy or girl — In theory, yes — there are two outcomes and we usually treat them as equally likely for simple models, though real-world data shows a slight difference.

4. Write the sample space and calculate the probability.

(i) Two coins tossed. S = {HH, HT, TH, TT}. P(at least one head) = 34.<

(ii) Ten cards numbered 1 to 10. Even numbers = {2, 4, 6, 8, 10}. P(even) = 510 = 12.<

(iii) A die rolled once. Numbers > 4 are {5, 6}. P = 26 = 13.<

(iv) Bag with 3 red, 2 blue, 1 green. Not red means blue or green (3 balls). P(not red) = 36 = 12.<

(v) Three coins tossed. S has 8 outcomes. Exactly two heads = {HHT, HTH, THH}. P = 38.

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

P(Strawberry) = 13.

6. A child has 2 shirts (red, blue) and 3 pants (jeans, khakis, shorts). List all possible outfits in a table.

Total outfits = 2 × 3 = 6 combinations.

7. Tyre replacement distances for 1,000 cases.

(i) P(less than 4000 km) = 201000 = 0.02.
(ii) P(between 4000 and 14000 km) = 210 + 3251000 = 5351000 = 0.535.
(iii) P(more than 14000 km) = 4451000 = 0.445.

8. The letters of the word PEACE are placed on cards. Leela draws one card without looking.

Letters: P, E, A, C, E. Total = 5 cards.
(i) P(P, E, or C) = 1 + 2 + 15 = 45 = 0.8.
(ii) P(not an E) = 35 = 0.6. (The non-E cards are P, A, C.)

9. A game of chance uses a spinner with numbers 1 to 8 as equally likely outcomes.

(i) P(8) = 18.<
(ii) Odd numbers are {1, 3, 5, 7}. P(odd) = 48 = 12.<
(iii) Numbers > 2 are {3, 4, 5, 6, 7, 8}. P = 68 = 34.<
(iv) Numbers < 9 are {1, 2, 3, 4, 5, 6, 7, 8}. P = 88 = 1 (certain).<
(v) Multiples of 3 are {3, 6}. P = 28 = 14.

10. A basket has 4 red balls and 5 blue balls. One ball is drawn and laid aside, then a second is drawn.

Total = 9 balls. Tree diagram branches:
First draw: Red (49) → Second: Red (38) or Blue (58).<
First draw: Blue (59) → Second: Red (48) or Blue (48).<

(i) P(Red then Blue) = 49 × 58 = 2072 = 518.<
(ii) P(2 Blue balls) = 59 × 48 = 2072 = 518.

11. I throw a pair of 6-sided dice. Write an event with probability 0 and an outcome with probability 1.

Event with probability 0: "The sum of the two dice is 1" (the lowest sum is 2).<
Outcome with probability 1: "The sum is a number from 2 to 12" (every possible throw gives a sum in this range).

12. Calculate the probabilities.

(i) Two dice rolled. Prime sums greater than 5 are 7 and 11.
Ways to get 7: 6 ways. Ways to get 11: 2 ways. Total favourable = 8. Total outcomes = 36.
P = 836 = 29.<

(ii) Bag: 4 red, 3 green, 2 blue. Two balls drawn without replacement.
P(both different colours) = 1 – P(both same).<
P(RR) = 49·<38 = 1272. P(GG) = 39·<28 = 672. P(BB) = 29·<18 = 272.<
P(same) = 2072 = 518. Therefore P(different) = 1 – 518 = 1318.<

(iii) Three coins tossed. First coin shows heads and exactly two heads in total.
Fix first as H. We need exactly one more H in the next two tosses: {HHT, HTH}.<
P = 28 = 14.<

(iv) Four-digit number from 1, 2, 3, 4 with no repetition. Total numbers = 4! = 24.
For the number to be even, the last digit must be 2 or 4.
If last is 2: arrange 1, 3, 4 in front → 6 ways.
If last is 4: arrange 1, 2, 3 in front → 6 ways.
Favourable = 12. P = 1224 = 12.<

(v) 3 multiple-choice questions, 4 options each, guessing exactly 2 correct.
Choose which 2 are correct: C(3,2) = 3 ways.
P = 3 × (14)² × (34) = 3 × 116 × 34 = 964 ≈ 0.141.

13. A box contains 4 balls numbered 1 to 4.

(i) With replacement tree: Each first branch (1,2,3,4) leads to second branches (1,2,3,4). Sample space size = 4 × 4 = 16.<
(ii) Without replacement tree: Each first branch leads to three remaining numbers. Sample space size = 4 × 3 = 12.<
(iii) Sizes are 16 and 12 respectively.

14. List the sample space for tossing a coin and drawing a card from 6 cards numbered 1 through 6.

S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}. n(S) = 12.

15. Three coins are tossed and the number of heads is recorded. Which list is a valid sample space?

The correct list is {0, 1, 2, 3} (option iv).<
(i) {1,2,3} fails because 0 heads is possible (TTT).<
(ii) {0,1,2} fails because 3 heads is possible (HHH).<
(iii) {0,1,2,3,4} fails because 4 heads is impossible with only 3 coins.

16. A dye is dropped at random on a 3 m × 2 m rectangular region. What is the probability it lands inside a circle of diameter 1 m?

Area of rectangle = 3 × 2 = 6 m².
Radius of circle = 0.5 m. Area of circle = π × (0.5)² = π4 m².
P = π/46 = π24.