Class 10 Mathematics Chapter 14 Probability — Formulas & Key Points

These are the foundational formulas and definitions you must memorise verbatim for CBSE board exams. Every probability problem in Class 10 Mathematics rests on the classical definition of probability applied to equally likely outcomes. The table below presents the theoretical probability formula, the experimental probability formula, and the relationship between complementary events. Note that the NCERT textbook emphasises equally likely outcomes — meaning each outcome has the same chance of occurring, such as each face of a fair die or each card in a well-shuffled deck.

Understanding precise terminology is critical for writing correct answers in board exams. The CBSE marking scheme awards marks for using standard language. Below are the definitions exactly as used in NCERT Class 10 Mathematics. Experiment means an operation which can produce some well-defined outcomes. Trial is a single performance of an experiment. An event is a collection of outcomes of an experiment; it is a subset of the sample space. Equally likely outcomes are outcomes that have the same theoretical probability of occurring. For example, when tossing a fair coin, heads and tails are equally likely. In contrast, in a loaded die, outcomes are not equally likely.

These properties form the logical backbone of probability. First, probability is always a number from 0 to 1 inclusive. Second, the probability of the entire sample space is 1 — something must happen. Third, if an event is impossible, its probability is 0. Fourth, the sum of probabilities of all mutually exclusive and exhaustive events equals 1. Mutually exclusive means the events cannot occur together; exhaustive means together they cover the entire sample space. In Class 10, you primarily work with simple events in finite sample spaces, so these rules simplify many calculations.

CBSE Class 10 Mathematics Chapter 14 explicitly distinguishes experimental probability (empirical) from theoretical probability (classical). Experimental probability is calculated after conducting trials: divide the number of times the event occurred by the total number of trials. Theoretical probability is calculated using logic and counting, assuming equally likely outcomes. As the number of trials increases, experimental probability tends to get closer to theoretical probability — this is the law of large numbers, though not explicitly named in Class 10 NCERT. Many board questions present data from experiments and ask for experimental probability, or they ask you to compare experimental and theoretical values.

A fair six-sided die is rolled once. We want to find the probability of getting a prime number. Step 1: Identify the sample space S = {1, 2, 3, 4, 5, 6}, so total outcomes = 6. Step 2: List outcomes favourable to the event E (getting a prime). Prime numbers in 1 to 6 are 2, 3, and 5, so E = {2, 3, 5} and favourable outcomes = 3. Step 3: Apply the formula P(E) = 3/6 = 1/2. Therefore, the probability of rolling a prime number is 1/2 or 0.5. Always simplify fractions to lowest terms in your final answer unless the question specifies otherwise. This type of single-event probability is the most common in CBSE board papers.

A card is drawn randomly from a well-shuffled standard deck of 52 cards. Find the probability that the card is (a) a king, (b) a red card, (c) not a spade. Step (a): There are 4 kings in the deck (one per suit). P(king) = 4/52 = 1/13. Step (b): Red cards are hearts and diamonds, 26 cards total. P(red) = 26/52 = 1/2. Step (c): Spades are 13 cards. P(spade) = 13/52 = 1/4, so P(not spade) = 1 − 1/4 = 3/4. Alternatively, non-spade cards = 52 − 13 = 39, so P(not spade) = 39/52 = 3/4. Both methods yield the same answer, but using the complement is often faster.

Two fair coins are tossed simultaneously. Find the probability of getting (a) exactly one head, (b) at least one head. Step 1: Sample space S = {HH, HT, TH, TT}, total outcomes = 4. Step (a): Exactly one head means either HT or TH, so favourable outcomes = 2. P(exactly one head) = 2/4 = 1/2. Step (b): At least one head means one or more heads: {HH, HT, TH}, favourable outcomes = 3. P(at least one head) = 3/4. Alternatively, use the complement: P(at least one head) = 1 − P(no heads) = 1 − P(TT) = 1 − 1/4 = 3/4. The complement method is especially useful when the event 'at least one' would require summing many cases.

Probability is one of the shortest chapters in Class 10 Mathematics, but small mistakes cost marks. Use the mnemonic 'FIT' to remember the classical formula: Favourable outcomes over Total outcomes (with equally likely outcomes). Remember 'ZERO to ONE' — probability is always between 0 and 1. For complementary events, think 'Opposite adds to 1': P(E) + P(not E) = 1. When you see words like 'at least one', immediately consider using the complement P(at least one) = 1 − P(none). For card problems, memorise: 52 cards, 4 suits, 13 ranks, 26 red (hearts + diamonds), 26 black (clubs + spades), 12 face cards. For a single die, remember prime numbers {2, 3, 5} and even numbers {2, 4, 6}.

Students lose easy marks in probability by making small notation or interpretation errors. First mistake: writing probability as a ratio instead of a simplified fraction. Always simplify 4/52 to 1/13. Second mistake: forgetting to count all outcomes. For two dice, sample space has 36 outcomes (6×6), not 12. Third mistake: confusing 'and' with 'or'. 'And' usually means multiplication (for independent events, not in Class 10 syllabus formally), but in counting outcomes, list each compound outcome separately. Fourth mistake: not recognising equally likely outcomes. For example, sum of two dice is NOT equally likely — sum 7 is more common than sum 2. Fifth mistake: writing probability greater than 1 or negative, which is impossible. Sixth mistake: misinterpreting 'at least' or 'at most'. 'At least 2' means 2 or more; 'at most 2' means 2 or fewer.

This table gives you ready-made probabilities for the most common scenarios tested in CBSE Class 10 board exams. Memorise these to save calculation time during the exam. For a single fair die, P(any specific number) = 1/6; P(even) = P(odd) = 1/2; P(prime) = 1/2; P(composite among 1-6) = 2/6 = 1/3 (composite numbers are 4 and 6; note 1 is neither prime nor composite). For a single fair coin, P(H) = P(T) = 1/2. For two coins, P(both heads) = 1/4, P(exactly one head) = 1/2, P(at least one head) = 3/4. For a standard 52-card deck, P(any specific rank like ace) = 4/52 = 1/13; P(any specific suit) = 13/52 = 1/4; P(red) = P(black) = 1/2; P(face card) = 12/52 = 3/13.

Use this box for final revision 15 minutes before the exam. Core formula: P(E) = (Number of favourable outcomes)/(Total number of equally likely outcomes). Range: 0 ≤ P(E) ≤ 1 always. Complementary: P(not E) = 1 − P(E). Sum rule: sum of probabilities of all elementary events = 1. Impossible event P = 0; certain event P = 1. Experimental P(E) = (trials where E occurred)/(total trials). Standard scenarios: Die (6 outcomes), Coin (2 outcomes), Two coins (4 outcomes), Deck (52 cards: 26 red, 26 black, 12 face, 4 of each rank), Two dice (36 outcomes). Keywords: at least one → use 1 − P(none); exactly → count directly. Common mistakes: probability > 1, not simplifying, miscounting outcomes. Write probability as simplified fraction or decimal (0 to 1).

Probability questions in CBSE board exams often test interpretation as much as calculation. Students struggle with wording like 'at least', 'at most', or 'exactly', and they lose marks by miscounting outcomes or forgetting to simplify. CBSETUTOR.ai offers a 24×7 AI tutor that you can access from your phone: snap a photo of any probability problem from your NCERT Class 10 Mathematics textbook or sample paper, and get an instant step-by-step solution with explanations in simple Indian English. The AI tutor highlights common errors — like treating non-equally-likely outcomes as equal, or forgetting the complement rule — and provides practice questions tailored to CBSE marking patterns. At a flat ₹999 per month for all classes 6-12, it is far more affordable than hiring a private tutor in any city. Start your 3-day free trial today and see how real-time doubt solving builds confidence and saves hours of confusion, especially in compact chapters like Probability that demand precision.