Probability is the branch of mathematics that measures the chance of an event occurring. In everyday language we say that something is "likely", "unlikely", or "certain", and probability makes these ideas precise by assigning a number between 0 and 1 to every event. A probability of 0 means an event is impossible, 1 means it is certain, and values in between reflect varying degrees of likelihood.
This chapter introduces the theoretical (classical) approach to probability, where all outcomes are assumed to be equally likely. For an experiment with a finite number of equally likely outcomes, the probability of an event E is the ratio of the number of favourable outcomes to the total number of outcomes. The chapter deals with experiments such as tossing coins, throwing dice, and drawing cards from a pack, which form the classic setting for probability problems.
Probability has applications in science, business, weather forecasting, insurance, and games. Understanding how to count favourable outcomes and how to reason about equally likely events builds logical thinking and is also the foundation for the more advanced probability theory studied in higher classes. This chapter is a reliable source of marks and is generally considered one of the most enjoyable topics in the syllabus.
An experiment whose outcome cannot be predicted with certainty in advance is called a random experiment. Examples include tossing a coin, throwing a die, and drawing a card from a shuffled deck.
Each possible result of the experiment is an outcome. The set of all possible outcomes is the sample space.
Outcomes are equally likely if each one has the same chance of occurring. For example, the outcomes of a fair die are equally likely.
An event is a subset of the sample space, i.e., a collection of one or more outcomes. For example, "getting an even number" when throwing a die is an event with outcomes 2, 4, 6.
The outcomes belonging to the event are called the favourable outcomes.
If an experiment has n equally likely outcomes, and m of them are favourable to an event E, then: P(E) = m/n
Example: When throwing a fair die, the probability of getting a 4 is: P(E) = 1/6, since there is one favourable outcome out of six.
Example: The probability of getting an even number on a die: Favourable outcomes are 2, 4, 6 (three outcomes). P(E) = 3/6 = 1/2.
The complement rule P(not E) = 1 - P(E) is extremely useful and is used in many problems where directly counting favourable outcomes is tedious.
Outcomes: H, T. P(H) = 1/2, P(T) = 1/2.
Outcomes: HH, HT, TH, TT (four equally likely outcomes). - P(2 heads) = 1/4. - P(exactly one head) = 2/4 = 1/2. - P(at least one head) = 3/4. - P(no head) = 1/4.
Outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT (eight equally likely outcomes). - P(all heads) = 1/8. - P(at least two heads) = 4/8 = 1/2. - P(exactly one head) = 3/8.
The total number of outcomes doubles with each additional coin.
Outcomes: 1, 2, 3, 4, 5, 6. - P(3) = 1/6. - P(odd number) = 3/6 = 1/2. - P(number greater than 4) = 2/6 = 1/3.
There are 6 x 6 = 36 equally likely outcomes, usually written as ordered pairs (a, b). - P(sum = 7) = 6/36 = 1/6, since the pairs (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) all give 7. - P(doublet) = 6/36 = 1/6. - P(sum greater than 9) = 6/36 = 1/6 (pairs summing to 10, 11, 12).
When listing outcomes of two dice, ordering matters, so (1, 2) and (2, 1) are counted separately.
A standard pack of 52 cards consists of 4 suits (spades, hearts, diamonds, clubs) with 13 cards in each suit. The suits spades and clubs are black (26 cards), while hearts and diamonds are red (26 cards). Each suit has one ace, one king, one queen, one jack (the face cards, 4 x 4 = 16 in total) and 9 numbered cards.
Examples: - P(drawing an ace) = 4/52 = 1/13. - P(drawing a king) = 4/52 = 1/13. - P(drawing a red card) = 26/52 = 1/2. - P(drawing a face card) = 12/52 = 3/13 (only J, Q, K counted, not aces). - P(drawing a spade) = 13/52 = 1/4.
Careful reading is essential: "face cards" usually means J, Q, K (12 cards), not including aces.
Many board questions apply probability to real situations, such as: - Probability that a student has a particular birthday. - Probability of picking a particular kind of bulb or pen from a batch. - Probability based on the number of outcomes of a spinner. - Probability of selecting a student of a particular age from a group.
These are solved by the same definition: favourable outcomes divided by total outcomes. The essential skill is to count the total and favourable outcomes correctly, whether they come from a given frequency table or from the geometry of the situation.
| Experiment | Total Outcomes | Example Event | Probability |
|---|---|---|---|
| One coin | 2 | Getting a head | 1/2 |
| Two coins | 4 | Exactly one head | 1/2 |
| Three coins | 8 | All heads | 1/8 |
| One die | 6 | Getting an even number | 1/2 |
| Two dice | 36 | Sum equal to 7 | 1/6 |
| Pack of cards | 52 | Drawing an ace | 1/13 |
| Rule | Statement |
|---|---|
| Range | 0 <= P(E) <= 1 |
| Impossible event | P(E) = 0 |
| Sure event | P(E) = 1 |
| Complement | P(not E) = 1 - P(E) |
| Definition | P(E) = favourable outcomes/total outcomes |
The safest method for any probability problem is to list the sample space completely before counting favourable outcomes. Consider the event of getting a sum of at least 10 when two dice are thrown. The sample space has 36 ordered pairs, and we count only those pairs whose sum is 10, 11, or 12. For a sum of 10, the pairs are (4, 6), (5, 5), and (6, 4); for a sum of 11, the pairs are (5, 6) and (6, 5); and for a sum of 12, there is only (6, 6). Altogether there are 6 favourable outcomes, so the probability is 6/36, which simplifies to 1/6. Writing the pairs out in order prevents the double counting and omission that confuse many students, and it makes the reasoning visible for full method marks.
The complement rule is a powerful shortcut when an event is awkward to count directly. For example, to find the probability of getting at least one head when three coins are tossed, we could count the seven favourable outcomes, but it is easier to find the probability of its complement, getting no head at all. Only the outcome TTT has no head, so the probability of the complement is 1/8, and therefore the probability of at least one head is 1 - 1/8 = 7/8. This approach is especially valuable for questions such as "at least one" or "at most one", where the direct count is long and error-prone but the complement is a single easy case.
Problems drawn from everyday contexts, such as selecting bulbs from a batch or students from a group, follow exactly the same definition: probability equals the number of favourable outcomes divided by the total number of outcomes. When data is given in a frequency table, the total frequency is the total number of outcomes and the frequency of the required category is the number of favourable outcomes. The only new skill is careful reading, because a phrase such as "a bulb that is not defective" may be easier to compute as one minus the probability of a defective bulb. A related idea is that the sum of the probabilities of all mutually exclusive outcomes in an experiment is always 1, which provides a useful check on any calculation. Practising the habit of stating the total and the favourable counts in words before writing the fraction is the single most reliable way to score full marks in probability.
Probability provides a rigorous framework for describing uncertainty, and this chapter introduces its fundamental language and rules through classic experiments with coins, dice, and cards. The definition of theoretical probability as the ratio of favourable to total outcomes, together with the properties and the complement rule, equips students to solve a wide variety of problems. Beyond the examination, probability underpins decision-making in science, finance, and everyday life. This chapter is both engaging and scoring, and it lays the essential groundwork for the deeper probabilistic reasoning students will encounter in higher mathematics.