What is Probability?

Probability is the branch of mathematics that measures how likely an event is to occur. It answers questions like: "What are the chances of getting a head when I toss a coin?" or "How likely is it to draw a king from a deck of cards?" Probability is used everywhere — in weather forecasting, insurance, games, elections and scientific experiments.

The probability of any event always lies between 0 and 1 (inclusive). A probability of 0 means the event is impossible. A probability of 1 means the event is certain. Everything else falls in between.

Classifying Events: Less Likely, Equally Likely, More Likely

Before studying exact probabilities, we first learn to judge whether an event is less likely, equally likely, or more likely based on common sense and the structure of the experiment. This helps build intuition before the formula is introduced.

Random Experiment, Trial and Sample Space

What is an Event?

Equally Likely Events

Understanding a Deck of 52 Playing Cards

A standard deck of playing cards has 52 cards split into 4 suits of 13 cards each. Each suit contains: Ace (A), 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack (J), Queen (Q) and King (K). The Jack, Queen and King are called face cards or court cards — there are 12 face cards in total (3 per suit).

Experimental (Empirical) Probability

Experimental probability is calculated by actually performing an experiment many times and recording the results. It is also called empirical probability. The more trials you conduct, the closer the experimental probability gets to the theoretical value.

The lesson demonstrates this with a classic coin-toss experiment. The table below shows how the relative frequency of getting a Head changes as the number of trials increases:

Theoretical (Classical) Probability

Example 1 — Tossing a Single Coin

Sample space: S = {H, T}  →  n(S) = 2

Event E: Getting a Head  →  E = {H}  →  n(E) = 1

Example 2 — Rolling a Fair Die

Sample space: S = {1, 2, 3, 4, 5, 6}  →  n(S) = 6

(Blue = even numbers)

Event A: Getting an even number  →  A = {2, 4, 6}  →  n(A) = 3

Worked Example — Two Coins Tossed Simultaneously

When two coins are tossed, the outcomes depend on what each coin shows. We list them as (Coin1, Coin2) pairs. This gives 4 equally likely outcomes.

Worked Example — Three Coins Tossed Simultaneously

Three coins give 2³ = 8 equally likely outcomes. Listing them systematically by working through all combinations of H and T:

Sure Events, Impossible Events and the Range of Probability

Experimental vs Theoretical Probability — Key Differences

Common Mistakes to Avoid

• Forgetting sample space size: Always list S first and count n(S) before attempting any probability calculation. For two coins, n(S) = 4 (not 3), because HT and TH are different outcomes.
• Treating prime and composite as equally likely: For a die, primes are {2,3,5} (3 numbers) and composites are {4,6} (only 2 numbers — note that 1 is neither prime nor composite). These are NOT equally likely.
• Ace as a face card: A is NOT a face card. Face cards are only Jack (J), Queen (Q) and King (K). There are 12 face cards, not 16.
• Misreading "at least" and "at most": "At least one head" means 1 or more heads — include all outcomes with ≥ 1 head. "At most two heads" means 2 or fewer heads — include all outcomes with ≤ 2 heads.
• Probability greater than 1: If your answer is > 1, you have made an error. Probability can never exceed 1.

What This Lesson Prepares You For

This introduction to probability builds the vocabulary and tools needed for the exercise questions in Exercise 14.1, where you apply all these concepts to solve structured probability problems involving coins, dice and card draws.

The theoretical probability framework introduced here expands significantly in Class 10 Probability, where you will encounter complementary events (P(not E) = 1 − P(E)), combined events, and more complex problems involving coloured balls and numbered cards. The deck-of-cards structure also appears repeatedly in Class 10 probability exercises.

For students looking to strengthen their foundation, revisiting Class 9 Statistics (Chapter 13) is useful — both chapters deal with collecting data and making predictions, and the concept of relative frequency in statistics directly connects to experimental probability here.