The Mathematics of Maybe: Probability Class 9 Notes

The Mathematics of Maybe: Introduction to Probability is the Class 9 chapter that turns "maybe" into a number. In this guide you will learn the probability scale, experimental and theoretical probability, sample spaces, tree diagrams, sampling and the law of large numbers, with each idea worked through and a set of important questions with model answers.

Randomness and random experiments

Many everyday situations have uncertain outcomes: will it rain today, which team will win the match? These are random events. We may know all the possible outcomes, but we cannot be 100% sure which one will happen in a single instance.

• Randomness: the result of an action, such as tossing a coin or rolling a die, cannot be predicted.
• Random experiment: an action that can be repeated under the same conditions, where the result may be different each time.

A cricket coin toss is fair because Heads and Tails have an equal chance, so neither captain can predict the result. Probability measures how likely an event is, much as we measure length or volume.

The probability scale

The probability of an event E, written P(E), always lies between 0 and 1: 0 ≤ P(E) ≤ 1.

A probability of 0.75 means a 75% chance. It is greater than 0.5, so the event is "more likely".

Experimental probability

Experimental probability (also called relative frequency) comes from real trials or past data:

Experimental probability = (number of times the event happened) ÷ (total number of trials)

• A paper cup is tossed 100 times and lands on its side 70 times: 70/100 = 0.7.
• A die is rolled 50 times and shows 4 exactly 8 times: 8/50 = 0.16, or 16%.
• A coin is tossed 20 times with 12 Heads: 12/20 = 3/5 = 0.6, or 60%.

Business, insurance and science use this method because it relies on evidence from experience. The answer is an estimate from that particular set of trials.

Theoretical probability

Theoretical probability needs no experiment. It comes from reasoning, assuming the coin or die is unbiased (fair): symmetrical, with no reason to favour one side.

P(E) = (number of favourable outcomes) ÷ (total number of possible outcomes)

Worked example 1: In the word PROBABILITY there are 11 letters, and B appears 2 times. P(B) = 2/11 ≈ 0.182, or about 18.2%.

Worked example 2: Probability of an even number on a fair die. Outcomes {1, 2, 3, 4, 5, 6}, total 6. Favourable {2, 4, 6}, which is 3. P(even) = 3/6 = 1/2 = 0.5.

Sample spaces and events

• Sample space (S): the complete list of all possible outcomes.
• Element: each single outcome in S.
• Sample size n(S): the number of elements.
• Event (E): a subset of S, one outcome or a group of outcomes. P(E) = n(E) ÷ n(S).

Toss two coins: S = {HH, HT, TH, TT}, n(S) = 4. The event "at least one head" is E = {HH, HT, TH}, so P(E) = 3/4.

Worked example (pairing): A fair offers 3 snacks (Samosa, Pakora, Bhaji) and 2 drinks (Chai, Lassi). Pair every snack with every drink: S = {(Samosa, Chai), (Samosa, Lassi), (Pakora, Chai), (Pakora, Lassi), (Bhaji, Chai), (Bhaji, Lassi)}, so n(S) = 3 × 2 = 6.

Tree diagrams

A tree diagram lists every outcome of a multi-step experiment. Each branch is one possible result of one step; following a path from start to end gives one final outcome. It makes sure no outcome is missed or repeated.

Tossing a coin twice:

1. Start.
2. First toss: two branches, H or T, each with probability 1/2.
3. Second toss: from H, branch to H or T; from T, branch to H or T.
4. Outcomes: HH, HT, TH, TT. Total = 4.

P(two heads) = 1/4 = 0.25, because only HH is favourable.

Sampling and the law of large numbers

Sampling means collecting data from a representative part of a population and using it to estimate the whole.

Worked example: 40% of a sample of 50 students like mangoes. For a school of 1500 students: 40% of 1500 = 0.4 × 1500 = 600 students. Larger, unbiased samples give better estimates.

The law of large numbers: as the number of trials increases, experimental probability gets closer and closer to theoretical probability. A coin might show 7 Heads in 10 tosses (0.7), but over 10,000 tosses the fraction of Heads will likely be very close to 0.5.

Remember this

• 0 ≤ P(E) ≤ 1; 0 is impossible, 1 is certain, 0.5 is an even chance.
• Experimental = what happened ÷ trials. Theoretical = favourable ÷ total possible.
• Write the sample space first, then count.
• Two-step experiments: multiply the choices (3 snacks × 2 drinks = 6).
• More trials bring experimental probability closer to theoretical.

Important questions with answers

1. Why is a coin toss a random experiment? We know the possible outcomes (Heads or Tails) but cannot predict which will occur in a given toss, and the toss can be repeated under the same conditions.

2. What is the probability of getting a prime number on a fair die? Primes on a die are {2, 3, 5}, which is 3 outcomes out of 6. P = 3/6 = 1/2.

3. A box has 5 green and 7 red balls. Find n(S) and P(green) for one draw. n(S) = 5 + 7 = 12. P(green) = 5/12.

4. Two coins are tossed. Find P(exactly one head). Favourable outcomes are HT and TH, which is 2 of 4. P = 2/4 = 1/2.

5. In the snack-and-drink example, what is the probability that a random pair includes Lassi? Pairs with Lassi: (Samosa, Lassi), (Pakora, Lassi), (Bhaji, Lassi), which is 3 of 6. P = 1/2.

6. Find the probability of picking the letter I from PROBABILITY. I appears 2 times among 11 letters. P(I) = 2/11.

7. A sample of 40 students shows 9 prefer sports. Estimate the number in a school of 800. 9/40 = 0.225. Then 0.225 × 800 = 180 students.

8. A coin gave 7 Heads in 10 tosses. Is it unfair? Not necessarily. Ten trials is a small number. By the law of large numbers, many more tosses should bring the fraction close to 0.5 if the coin is fair.

Common mistakes to avoid

• Giving a probability above 1 or below 0; check your answer lies in this range.
• Mixing up experimental and theoretical probability.
• Listing HT and TH as one outcome; they are different.
• Forgetting to simplify: 12/20 = 3/5.

For more practice with sample spaces and tree diagrams, study this chapter with Joy on Learnijoy.