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CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability — Notes

CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability opens the door to understanding uncertainty through numbers. Probability is the mathematical tool that measures how likely an event is to occur—whether you are tossing a coin, rolling a die, or predicting tomorrow's weather. This chapter grounds students in two complementary methods: experimental probability, built from real observations and data collection, and theoretical probability, derived from logical reasoning about equally likely outcomes. By the end of this chapter, students will confidently calculate probabilities, interpret sample spaces, distinguish outcomes from events, and use tree diagrams to analyse multi-step experiments. These skills form the bedrock for advanced topics in statistics, data science, and decision theory in higher classes.

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Key takeaways

  • ✓CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability teaches both experimental and theoretical approaches to measuring likelihood.
  • ✓A random experiment has known possible outcomes but unpredictable results; the sample space is the complete list of all possible outcomes.
  • ✓Experimental probability equals the number of times an event occurred divided by the total number of trials, reflecting real-world data.
  • ✓Theoretical probability equals the number of favourable outcomes divided by the total number of equally likely outcomes, assuming fairness.
  • ✓Probability values always lie between 0 (impossible event) and 1 (certain event), with 0.5 representing an equally likely outcome.
  • ✓Tree diagrams visually map all outcomes in multi-step experiments and help calculate probabilities by multiplying along branches for independent events.
  • ✓The Law of Large Numbers states that as the number of trials increases, experimental probability converges toward theoretical probability.

Understanding Random Experiments and the Nature of Randomness

A random experiment is any process or action where you know all the possible outcomes in advance, but you cannot predict which specific outcome will occur in any single trial. The defining feature is unpredictability: no amount of skill, knowledge, or prior observation allows you to guarantee the result beforehand. Classic examples include tossing a fair coin (outcomes: heads or tails), rolling a standard six-sided die (outcomes: 1, 2, 3, 4, 5, 6), or drawing a card from a shuffled deck. Even in everyday contexts—such as selecting a student randomly from a class list or picking a sweet blindly from a mixed bag—the mechanism is fair and unbiased, yet the outcome remains unknown until the experiment is performed. Randomness is not chaos; it is structured unpredictability. The sample space is well-defined, but which element will emerge is genuinely uncertain. This concept is central to CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability, as it establishes the foundation for all probability calculations. Without true randomness, probability would collapse into certainty (P = 1) or impossibility (P = 0), with no meaningful middle ground.
  • A random experiment has a complete, known set of possible outcomes but an unpredictable result for any single trial.
  • Examples: coin toss (H or T), die roll (1–6), drawing a card from a shuffled deck, selecting a name from a hat.
  • Predictable events like 'the sun will rise tomorrow' are not random experiments—they have probability 1 (certain).
  • Randomness is essential for probability: if you could predict the outcome, probability would always be 0 or 1.

Sample Space, Outcomes, and Sample Size in Probability Experiments

The sample space, denoted S, is the exhaustive list of all possible outcomes of a random experiment. Each individual result in this list is called an outcome or element. The sample size, denoted n(S), is the count of distinct outcomes in the sample space. For example, when you toss a single fair coin, the sample space is S = {H, T} and n(S) = 2. When you roll a standard die, S = {1, 2, 3, 4, 5, 6} and n(S) = 6. For two coin tosses, outcomes must account for sequence: S = {HH, HT, TH, TT}, so n(S) = 4. Correctly identifying the sample space is the first and most critical step in any probability problem. A common mistake is undercounting outcomes (e.g., forgetting that HT and TH are different when order matters) or overcounting (listing the same outcome twice). In CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability, students learn to systematically enumerate outcomes using lists, tables, or tree diagrams. The sample space must be complete (every possibility included), mutually exclusive (no overlap), and appropriate to the question (if you care about rain intensity, don't just use {Rain, No Rain}—expand it to {No Rain, Drizzle, Light Rain, Heavy Rain}). Understanding sample space is non-negotiable because both experimental and theoretical probability depend on knowing the denominator: the total number of possible outcomes.
  • Sample space S is the set of all possible outcomes; sample size n(S) counts them.
  • One coin: S = {H, T}, n(S) = 2. One die: S = {1,2,3,4,5,6}, n(S) = 6. Two coins: S = {HH, HT, TH, TT}, n(S) = 4.
  • Outcomes must be distinct and exhaustive—missing an outcome leads to incorrect probability.
  • Use tree diagrams or systematic lists to ensure you capture every outcome, especially in multi-step experiments.

Events and Favourable Outcomes: Defining What You Measure

An event is any subset of the sample space—essentially, it is a collection of one or more outcomes that satisfy a specific condition or criterion. For instance, if you roll a die, the event 'rolling an even number' corresponds to the subset {2, 4, 6}. An event can be simple (containing exactly one outcome, such as 'rolling a 3') or compound (containing multiple outcomes, such as 'rolling a number greater than 4', which is {5, 6}). Favourable outcomes are the outcomes in the sample space that belong to the event of interest. The number of favourable outcomes determines the numerator in the probability formula. In CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability, distinguishing between an outcome (a single result) and an event (a set of results) is crucial. The entire sample space is called the certain event (P = 1), because at least one outcome must occur. The empty set (no outcomes) is the impossible event (P = 0). Events can overlap: for example, 'rolling an even number' and 'rolling a number greater than 3' both include the outcomes 4 and 6. Understanding events allows you to translate a worded question into a mathematical set, which you can then count and use in probability calculations.
  • An event E is a subset of the sample space; it groups outcomes satisfying a condition.
  • Simple event: one outcome (e.g., rolling exactly 3). Compound event: multiple outcomes (e.g., rolling even: {2,4,6}).
  • Favourable outcomes are the outcomes in S that belong to event E.
  • Certain event: S itself (P=1). Impossible event: empty set (P=0).

Experimental (Empirical) Probability: Learning from Real Data

Experimental probability, also called empirical probability, is the likelihood of an event based on actual trials and observed frequencies. It is calculated as the ratio of the number of times the event occurred to the total number of trials performed. The formula is: Experimental Probability = (Number of times event occurred) / (Total number of trials). For example, if you toss a coin 60 times and observe heads 34 times, the experimental probability of heads is 34/60 ≈ 0.567 or 56.7%. This method relies on real data, not assumptions about fairness. It is especially valuable when the situation is too complex for theoretical analysis or when bias might be present. A coin might be bent, a die might be loaded, or a spinner might have unequal friction—experimental probability reveals the actual behaviour. In CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability, students perform experiments such as tossing coins, rolling dice, or drawing objects from a bag, recording results in frequency tables, and computing probabilities. The Law of Large Numbers states that as the number of trials increases, experimental probability tends to converge toward theoretical probability (if outcomes are truly equally likely). With 10 trials, you might see wild fluctuations; with 1000 trials, the pattern stabilises. This is why insurance companies, weather services, and quality control departments rely on large datasets to estimate probabilities accurately.
  • Experimental probability = (occurrences of event) / (total trials).
  • Based on real data, not assumptions—useful when fairness is uncertain or outcomes are not equally likely.
  • Example: Roll a die 50 times, get a 4 eight times → P(4) = 8/50 = 0.16 (16%).
  • Law of Large Numbers: more trials → experimental probability approaches theoretical probability.

Theoretical Probability: Reasoning About Equally Likely Outcomes

Theoretical probability is the probability calculated by logical reasoning under the assumption that all outcomes in the sample space are equally likely. The formula is: Theoretical Probability P(Event) = (Number of favourable outcomes) / (Number of possible outcomes). This approach requires no experiments—just counting. For instance, when rolling a fair six-sided die, each outcome (1, 2, 3, 4, 5, 6) has equal chance. To find P(rolling a number greater than 4), identify favourable outcomes {5, 6} (2 outcomes) and divide by the total 6 outcomes: P = 2/6 = 1/3 ≈ 0.333 or 33.3%. Theoretical probability is powerful because it is fast and precise, but it depends entirely on the assumption of fairness. A biased die, a weighted coin, or a shuffled deck with marked cards violates this assumption, making theoretical probability inaccurate. In CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability, students apply this method to coins, dice, cards, and spinners, always checking that outcomes are genuinely equally likely. Theoretical probability represents the ideal or expected behaviour in a perfectly fair system. In practice, experimental probability may differ slightly due to random variation, but over many trials, the two should align closely (Law of Large Numbers).
  • Theoretical probability = (favourable outcomes) / (total outcomes), assuming equal likelihood.
  • Requires no experiments—pure logic and counting.
  • Valid only when outcomes are truly equally likely (fair coin, unbiased die, well-shuffled deck).
  • Example: P(drawing a heart from a standard 52-card deck) = 13/52 = 1/4 = 0.25 (25%).

The Probability Scale: From Impossible to Certain

Probability is always expressed as a number between 0 and 1 (inclusive), or equivalently as a percentage between 0% and 100%. This range is called the probability scale. At P = 0, the event is impossible—it cannot happen under any circumstances (e.g., rolling a 7 on a standard die). At P = 1, the event is certain—it will definitely occur (e.g., drawing a red or black card from a standard deck, since all cards are one or the other). At P = 0.5 (50%), the event is equally likely to happen or not happen (e.g., tossing a fair coin and getting heads). Values between 0 and 0.5 indicate unlikely but possible events; values between 0.5 and 1 indicate likely but not guaranteed events. For example, if the probability of rain tomorrow is 0.7 (70%), rain is more likely than not, and you would probably carry an umbrella. If P = 0.1 (10%), rain is unlikely but not impossible. CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability emphasises that probabilities can never be negative (no such thing as less than impossible) and never exceed 1 (no such thing as more than certain). Understanding this scale helps students interpret probabilities intuitively and make informed decisions based on likelihood.
  • Probability scale: 0 ≤ P ≤ 1 (or 0% to 100%).
  • P = 0: impossible. P = 0.5: even chance. P = 1: certain.
  • 0 < P < 0.5: unlikely but possible. 0.5 < P < 1: likely but not certain.
  • Example: P(rain) = 0.7 means 70% chance—more likely than not.

Tree Diagrams: Visualising Multi-Step Experiments

A tree diagram is a graphical tool that maps out all possible outcomes of a multi-step random experiment. Each branch represents one possible result at a particular stage, and branches split to show the next set of possibilities. By following a path from the root (start) to a leaf (end), you trace one complete outcome. Tree diagrams are invaluable for counting outcomes systematically, especially when the sample space is large or complex. For example, tossing a coin twice produces a tree: start with two branches (H or T), then from each of those, two more branches (H or T again), yielding four paths: HH, HT, TH, TT. This confirms n(S) = 4. Tree diagrams also support probability calculations. If the stages are independent (the result of one stage does not affect the next), you multiply probabilities along a path. For instance, P(HH) = P(H on toss 1) × P(H on toss 2) = (1/2) × (1/2) = 1/4. In CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability, students use tree diagrams to solve problems involving two coins, a coin and a die, or drawing objects with replacement. The visual structure reduces errors and clarifies the logic of sequential experiments.
  • Tree diagram: a branching chart showing all outcomes step-by-step.
  • Each path from root to leaf = one outcome. Number of paths = sample size.
  • Useful for multi-step experiments (e.g., two coin tosses, drawing with replacement).
  • For independent events, multiply probabilities along a path: P(A then B) = P(A) × P(B).

Real-Life Probability, Statistical Probability, and Sampling

In real-world contexts, calculating theoretical probability for every event is often impractical or impossible due to complexity, cost, or sheer scale. Instead, we use statistical probability: collecting data from a representative sample and using that data to estimate probabilities for the entire population. A sample is a smaller, manageable subset of the population (the full group of interest). For example, a tyre manufacturer wanting to know what fraction of tyres last beyond 14,000 km cannot test all 1,000,000 tyres produced annually. Instead, they test 1000 tyres (the sample). If 445 of those lasted over 14,000 km, the estimated probability is 445/1000 = 0.445 or 44.5%. They can then predict that approximately 0.445 × 1,000,000 = 445,000 tyres will meet that standard. The accuracy of this estimate depends on sample size (bigger is usually better) and representativeness (the sample must reflect the diversity of the population). In CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability, students encounter examples like surveying classmates about favourite sports or colours, then scaling up the observed frequency to estimate school-wide preferences. This technique underpins opinion polls, medical trials, quality control, and market research. A key insight is that statistical probability is never perfectly exact—it is an estimate with some margin of error—but with careful sampling, it becomes highly reliable and actionable.
  • Statistical probability: estimate from a sample, then generalise to the population.
  • Sample = subset of population. Larger, more representative samples → better estimates.
  • Example: Test 1000 tyres, 445 last >14,000 km → P ≈ 0.445. Predict 445,000 of 1,000,000 will last.
  • Used in polls, medical studies, quality control—essential when testing the entire population is impractical.

The Law of Large Numbers and Convergence of Probabilities

The Law of Large Numbers is a fundamental principle stating that as the number of trials in a random experiment increases, the experimental probability of an event tends to converge toward its theoretical probability. In simple terms: more trials mean more reliable results. Suppose the theoretical probability of rolling a 6 on a fair die is 1/6 ≈ 0.1667. If you roll the die 10 times, you might get a 6 only once (experimental P = 1/10 = 0.1), or you might get it three times (experimental P = 3/10 = 0.3). These fluctuations are normal with small samples. But if you roll 1000 times, you will almost certainly observe close to 167 sixes, giving experimental P ≈ 167/1000 = 0.167, remarkably close to 1/6. This convergence does not guarantee that every small batch will match theory—short-term randomness persists—but over the long run, patterns stabilise. In CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability, this principle reassures students that discrepancies between experimental and theoretical values are expected initially and will diminish as they collect more data. The Law of Large Numbers also explains why casinos, insurance companies, and weather forecasters can make reliable predictions: they rely on enormous datasets where random noise averages out.
  • Law of Large Numbers: more trials → experimental probability approaches theoretical probability.
  • Small samples show high variability; large samples stabilise around the true value.
  • Example: 10 coin tosses might give 7 heads (70%), but 1000 tosses will yield close to 500 heads (50%).
  • Underpins all statistical inference: large datasets smooth out random fluctuations.

Complementary Events and the Sum-to-One Rule

For any event E, the complement of E (denoted E' or 'not E') consists of all outcomes in the sample space that do not belong to E. A fundamental rule in probability is that the sum of the probabilities of an event and its complement is always 1: P(E) + P(not E) = 1. This follows logically because one of two things must happen: either E occurs or it does not, with no third option. For instance, if P(rain tomorrow) = 0.3 (30%), then P(no rain tomorrow) = 1 − 0.3 = 0.7 (70%). This rule is extremely useful for simplifying calculations. Sometimes it is easier to compute the probability of the complement first, then subtract from 1. For example, finding P(at least one head in two coin tosses) directly requires counting {HH, HT, TH}—three outcomes. Alternatively, the complement is 'no heads at all' = {TT}, which is just one outcome, so P(no heads) = 1/4, and thus P(at least one head) = 1 − 1/4 = 3/4. In CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability, students apply this rule to solve problems more efficiently and to check their answers (if you calculated P(E) and P(not E), they must add to 1; if not, you made a mistake).
  • Complementary events: E and 'not E' cover the entire sample space with no overlap.
  • Rule: P(E) + P(not E) = 1, so P(not E) = 1 − P(E).
  • Useful shortcut: sometimes computing P(not E) is easier, then subtract from 1.
  • Example: P(drawing a non-spade from a deck) = 1 − P(spade) = 1 − 13/52 = 39/52 = 3/4.

Common Pitfalls and the Gambler's Fallacy

A frequent mistake in probability reasoning is the Gambler's Fallacy: the mistaken belief that past outcomes of independent random events influence future outcomes. For example, if a fair coin has landed heads five times in a row, some people believe tails is 'due' and more likely on the next toss. This is false. Each coin toss is independent; the coin has no memory. The probability of heads on the sixth toss remains exactly 1/2, regardless of previous results. Similarly, if a roulette wheel has landed on red several times consecutively, that does not make black more likely next time (assuming a fair wheel). The Gambler's Fallacy arises from a misunderstanding of the Law of Large Numbers, which applies over the long run, not in the short term. Another common error is confusing experimental and theoretical probability. Theoretical probability assumes fairness and equal likelihood; if the die is loaded or the coin is bent, theoretical calculations are invalid. Students must also avoid the mistake of treating non-equally-likely outcomes as if they were equal. For instance, the sample space for sum of two dice is not {2, 3, 4, …, 12} with equal probabilities—some sums (like 7) occur far more often than others (like 2 or 12). CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability teaches students to identify and avoid these logical traps through careful reasoning and attention to independence and fairness assumptions.
  • Gambler's Fallacy: believing past results affect future independent trials (they don't).
  • Each trial is independent—a coin or die has no memory.
  • Example: Five heads in a row does not make tails more likely on the sixth toss; P(H) still = 0.5.
  • Other pitfalls: assuming unequal outcomes are equal, ignoring bias, confusing experimental vs. theoretical probability.

Worked Example: Finding Theoretical Probability from a Word

Problem: A letter is chosen at random from the word 'PROBABILITY'. What is the probability that the letter chosen is the letter I? Solution: Step 1 — Identify the sample space. The word PROBABILITY contains 11 letters: P-R-O-B-A-B-I-L-I-T-Y. Hence the sample space S has n(S) = 11 outcomes. Step 2 — Identify the event and count favourable outcomes. Event E = 'choosing the letter I'. The letter I appears twice in PROBABILITY (positions 7 and 9). Number of favourable outcomes = 2. Step 3 — Apply the theoretical probability formula. P(I) = (Number of favourable outcomes) / (Total number of outcomes) = 2/11 ≈ 0.1818 or about 18.2%. Answer: The probability of choosing the letter I is 2/11 or approximately 0.182. This example from CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability demonstrates the importance of carefully counting both the total sample space and the favourable outcomes. Mistakes often occur when students miscount repeated letters or overlook case sensitivity (though typically problems treat uppercase and lowercase as the same unless stated otherwise).

Worked Example: Experimental Probability and Scaling to Population

Problem: A school conducts a survey of 50 students to find their favourite sport. Results: 18 Cricket, 14 Football, 10 Badminton, 8 Basketball. (i) Find the experimental probability that a randomly selected student prefers Football. (ii) If the school has 800 students, estimate how many prefer Football. Solution: Step 1 — Compute experimental probability. Total trials (students surveyed) = 50. Event: Prefers Football. Occurrences = 14. Experimental P(Football) = 14/50 = 7/25 = 0.28 or 28%. Step 2 — Scale to the full population. Total students = 800. Estimated number who prefer Football = P(Football) × Total students = 0.28 × 800 = 224 students. Answer: (i) P(Football) = 0.28 or 28%. (ii) Approximately 224 students out of 800 prefer Football. This example illustrates statistical probability and sampling, core ideas in CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability. The accuracy of the estimate depends on how representative the 50-student sample is of the entire school. A larger or more diverse sample would improve reliability.

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Frequently asked questions

What is the main difference between experimental and theoretical probability in CBSE Class 9 Mathematics Chapter 7?+
Experimental probability is calculated from actual trials and observed data (occurrences ÷ trials), reflecting real-world results. Theoretical probability is computed by reasoning about equally likely outcomes (favourable ÷ total), assuming a fair, unbiased setup. Experimental values may vary due to chance, especially with few trials, but converge toward theoretical values as trials increase (Law of Large Numbers).
How do I know if outcomes in a probability experiment are equally likely?+
Outcomes are equally likely if the experiment is fair and unbiased, with no physical or logical reason for one outcome to occur more often than another. For example, a fair coin, a balanced die, and a well-shuffled deck all produce equally likely outcomes. If a coin is bent, a die is loaded, or a deck is marked, outcomes are not equally likely, and you must rely on experimental data rather than theoretical formulas.
Why do experimental probability values differ from theoretical probability even for a fair coin or die?+
Random variation causes short-term fluctuations. With only 10 or 20 trials, you might observe heads 70% of the time purely by chance, even though theoretical P(heads) = 50%. As you perform hundreds or thousands of trials, experimental probability stabilises around the theoretical value due to the Law of Large Numbers. Small-sample deviations are normal and expected.
What is a sample space and why is it so important in CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability?+
The sample space S is the complete set of all possible outcomes of a random experiment. It is the denominator in every probability formula (both experimental and theoretical). If you miss an outcome or count duplicates, your probability calculation will be wrong. Always list or diagram the sample space systematically before computing probabilities.
Can probability be greater than 1 or less than 0?+
No. Probability is always between 0 and 1 (inclusive). P = 0 means impossible; P = 1 means certain. A value less than 0 or greater than 1 indicates a calculation error—recheck your numerator and denominator, and verify that the event is a subset of the sample space.
What is the Gambler's Fallacy and how do I avoid it?+
The Gambler's Fallacy is the incorrect belief that past results of independent random events affect future outcomes. For example, thinking that after five heads in a row, tails is 'due' on the next coin toss. Each toss is independent; the coin has no memory. Always remember: past results do not change the probability of future independent trials.
How do tree diagrams help solve probability problems in CBSE Class 9 Mathematics Chapter 7?+
Tree diagrams visually map all possible outcomes step-by-step in multi-stage experiments. Each branch represents one possibility at a stage; paths from root to leaf represent complete outcomes. This prevents missing outcomes and clarifies the sample space. For independent events, you can also multiply probabilities along a path to find the probability of that specific sequence.
If I survey 40 students and 15 prefer cricket, can I say exactly 375 students prefer cricket in a school of 1000?+
Not exactly—375 is an estimate based on the sample. Statistical probability gives a prediction, not a guarantee. The true number might be 370 or 380 due to sampling variability. Larger, more representative samples yield more accurate estimates, but all sample-based predictions have some margin of error.
What does it mean when CBSE Class 9 Mathematics Chapter 7 says 'equally likely outcomes'?+
Equally likely means each outcome has the same probability of occurring. For a fair six-sided die, each face (1,2,3,4,5,6) has probability 1/6. If outcomes are not equally likely (e.g., a biased die), you cannot use the simple theoretical formula; instead, perform experiments to measure actual frequencies.
How many trials should I perform to get reliable experimental probability?+
More is better. With 10–20 trials, expect high variability. With 100+ trials, results become more stable. With 1000+ trials, experimental probability closely approximates theoretical probability (if outcomes are truly equally likely). The Law of Large Numbers guarantees convergence as trial count increases, but there is no fixed 'magic number'—just aim for as many as practical.
My child's school uses a different textbook for Class 9 Mathematics. Will CBSE Class 9 Mathematics Chapter 7 notes still help?+
Yes. All CBSE-affiliated schools follow the NCERT syllabus framework for Class 9, even if they use supplementary or private publishers. The core concepts—random experiments, sample space, experimental vs. theoretical probability, tree diagrams—are identical across books. NCERT is the official reference, so mastering CBSE Class 9 Mathematics Chapter 7 The Mathematics of Maybe: Introduction to Probability using NCERT-based notes ensures your child covers every competency tested in board-pattern assessments.
What is the relationship between probability and percentage?+
Probability as a decimal (e.g., 0.25) can be converted to percentage by multiplying by 100 (0.25 × 100 = 25%). Both represent the same likelihood. Saying 'probability is 0.6' and 'there is a 60% chance' mean exactly the same thing. Use whichever form the question requests or whichever is clearer in context.

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