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Empirical Probability and Types of Events

🎓 Class 9 Mathematics CBSE Theory Ch 7 — The Mathematics of Maybe: Introduction to Probability ⏱ ~40 min
🌐 Language: [gtranslate]

This MCQ module is based on: Empirical Probability and Types of Events

This mathematics assessment will be based on: Empirical Probability and Types of Events
Targeting Class 9 level in Probability, with Intermediate difficulty.

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7.6 Three Views of Probability

Mathematicians and scientists use three different but related approaches to assigning probabilities. Each one is useful in different situations.

📐
Classical (Theoretical)
Counts equally likely outcomes. Used for fair coins, dice and cards. \[ P = \frac{n(E)}{n(S)} \]
📊
Empirical? (Experimental)
Computed from observed data after performing many trials. \[ P = \frac{\text{No. of trials with } E}{\text{Total trials}} \]
🧠
Subjective?
Based on personal judgement, experience or expert opinion when data is incomplete. Used in weather forecasting, medical risk, market analysis.

7.7 Empirical Probability — Learning from Data

Definition — Empirical (Experimental) Probability
Suppose a random experiment is repeated n times under identical conditions, and an event E happens m times. Then the empirical probability of E is:

\[ P(E) \;=\; \dfrac{m}{n} \;=\; \dfrac{\text{Number of trials in which } E \text{ occurred}}{\text{Total number of trials}} \]

Important — Law of Large Numbers (informal)
As the number of trials becomes very large, the empirical probability of an event tends to settle close to its classical (theoretical) probability. This is why we trust averages over thousands of trials more than averages over ten.

Example 8 — A coin tossed many times

A coin was tossed 100 times and Head appeared 53 times. The empirical probability of Head is

\(P(H) = \dfrac{53}{100} = 0.53\).

The classical probability is 0.5. The two values are close because 100 is large enough.

P 0.5 0 Number of trials (n) Theoretical 0.5 Empirical P(H)
Fig. 7.5 — As the number of coin tosses grows, the empirical probability oscillates and settles near the theoretical 0.5.

Example 9 — Survey data

A class of 50 students recorded their preferred sport. The data is shown below.

SportCricketFootballBadmintonOtherTotal
No. of students22149550

If a student is chosen at random, what is the empirical probability that the student prefers (i) Cricket, (ii) Football or Badminton, (iii) something other than Cricket?

Solution
n = 50.
(i) \(P(\text{Cricket}) = 22/50 = 0.44\).
(ii) Favourable = 14 + 9 = 23. \(P = 23/50 = 0.46\).
(iii) Not cricket = 50 - 22 = 28. \(P = 28/50 = 0.56\). Equivalently, \(1 - 0.44 = 0.56\). ✓

Example 10 — Birthdays in a year

Out of a sample of 600 newborns recorded at a hospital in 2024, 312 were boys. The empirical probability of a newborn being a boy is

\(P(\text{boy}) = \dfrac{312}{600} = 0.52\).

Classical reasoning suggests \(P = 0.5\), but real-world data often shows a slight skew toward boys at birth.

7.8 Classical vs Empirical — When Do They Match?

Key Comparison
Classical: uses logic + symmetry (need equally likely outcomes). Fixed value, computed before any experiment.
Empirical: uses observation. Changes from one set of trials to another but converges to the classical value when the experiment is fair and trials are many.
FeatureClassicalEmpirical
SourceReasoning about equally likely outcomesRepeated experiment or recorded data
Needs data?NoYes
Fixed value?YesChanges with each experiment
When to useCoins, dice, cards (fair)Real-world events, biased dice, opinion surveys

Example 11 — A "biased" coin

A coin is suspected of being biased. It is tossed 200 times and Head appears 124 times. Find the empirical probability of Head and tail. Is the coin biased?

Solution
\(P(H) = 124/200 = 0.62\). \(P(T) = 76/200 = 0.38\).
Classical theory predicts 0.5 each. Since 0.62 differs noticeably from 0.5 over a large sample of 200 trials, the coin appears biased toward Heads. (Statisticians use formal hypothesis tests in higher classes to be precise.)

7.9 Subjective Probability — When Trials Are Not Possible

Sometimes we cannot toss, draw or observe an event many times. We may need to assign a probability based on judgement.

  • "There is a 70% chance India will win tomorrow's match." — sports analyst.
  • "The probability that this patient has the disease, given the symptoms, is about 30%." — doctor.
  • "I'd say there's a 1 in 5 chance the price will rise next month." — economist.

These are subjective probabilities: they obey the same rules (lying between 0 and 1, complement summing to 1) but are based on belief rather than counts.

Historical Note
The mathematical theory of probability grew from Pascal and Fermat's 1654 letters about a gambling dispute. In India, much earlier, the Mahabharata story of Yudhishthira and the dice game in Sabha Parva (c. 400 BCE) shows that ideas of chance and fairness were thought about long before any formal theory existed.

7.10 Worked Examples — Mixed Practice

Example 12 — Two dice

Two dice are rolled together. What is the probability that the sum is (i) 7, (ii) at least 10, (iii) a prime number?

Sample space — pairs (a, b) for two dice b\a 123456 123456 234567 345678 456789 5678910 67891011 789101112 Red = sum 7 (6 cells); Green = sum ≥ 10 (6 cells)
Fig. 7.6 — All 36 outcomes when two dice are rolled, with sums highlighted.
Solution
n(S) = 6 × 6 = 36.
(i) Sum 7: pairs (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6 cases. \(P = 6/36 = 1/6\).
(ii) Sum ≥ 10: (4,6),(5,5),(6,4),(5,6),(6,5),(6,6) → 6 cases. \(P = 6/36 = 1/6\).
(iii) Prime sums in 2–12 are 2, 3, 5, 7, 11. Counts: 2→1, 3→2, 5→4, 7→6, 11→2. Total = 15. \(P = 15/36 = 5/12\).

Example 13 — Spinner

A spinner is divided into 8 equal sectors numbered 1–8. Find the probability that the spinner stops on (i) an odd number, (ii) a multiple of 3, (iii) a number greater than 5.

1234 5678
Fig. 7.7 — Spinner with 8 equally likely sectors numbered 1–8.
Solution
n(S) = 8.
(i) Odd = {1, 3, 5, 7}, n = 4. \(P = 4/8 = 1/2\).
(ii) Multiples of 3 = {3, 6}, n = 2. \(P = 2/8 = 1/4\).
(iii) Greater than 5 = {6, 7, 8}, n = 3. \(P = 3/8\).
Activity 7.2 — Convergence of Empirical Probability
L4 Analyse
Materials: A drawing pin (or thumbtack), notebook with a 5-column table, pen.
Predict: If you drop a drawing pin many times, what fraction will land "point up"? Write your guess (e.g., 0.6).
  1. Drop the pin from a height of about 30 cm onto a flat surface. Record whether it lands "point up" (U) or "point sideways" (S).
  2. Repeat for 20 trials. Count m = number of "U"s. Compute \(P_{20} = m/20\).
  3. Repeat for 50 trials in total. Compute \(P_{50}\). Then 100 trials → \(P_{100}\).
  4. Plot the values \(P_{20}, P_{50}, P_{100}\) on a graph.
  5. Compare with your predicted value. What do you observe about how \(P_n\) changes as n increases?

Different drawing pins have different "point-up" probabilities depending on shape and weight, but classes typically find values stabilising around 0.55–0.7 after 100 trials. Crucially, students see that the value jumps around at small n but settles down at large n — illustrating the law of large numbers.

Competency-Based Questions

Scenario: A weather station in Mumbai recorded the daily weather for 30 days in June. The records are: Sunny — 6 days, Cloudy (no rain) — 9 days, Light rain — 11 days, Heavy rain — 4 days.
Q1. What is the empirical probability that a randomly selected June day was rainy (light or heavy)?
L3 Apply
  • (a) 11/30
  • (b) 4/30
  • (c) 15/30 = 1/2
  • (d) 6/30
Answer: (c) 1/2. Rainy = 11 + 4 = 15. \(P = 15/30 = 1/2\).
Q2. A travel blogger writes, "Based on this data, you have a 50% chance of rain any day of the year in Mumbai." Analyse what is wrong with this claim.
L4 Analyse
Answer: The 0.5 figure is from June data only. June is a monsoon month in Mumbai. Generalising to "any day of the year" ignores seasonal variation: November–February are mostly dry. Empirical probability is only valid for the conditions and population it was measured in.
Q3. The classical model of "rain or no rain" with two outcomes would give P(rain) = 1/2 by symmetry. Evaluate whether this classical reasoning is appropriate for predicting Mumbai weather.
L5 Evaluate
Answer: No. Classical probability requires equally likely outcomes. Rain and no-rain are not symmetric — they depend on season, geography and atmospheric conditions. The "fair coin" model gives a coincidentally close 1/2 here only because June 2024 happened to be balanced. Empirical or subjective approaches are far more appropriate for weather.
Q4. Design a simple data-collection plan that a school weather club could run for one year so that they can compute month-wise empirical probabilities of rain. Describe what to record, how often, and how to compute the probabilities at the end.
L6 Create
One possible plan: (1) Each day, record date, max temp, and category R (rain ≥ 0.5 mm) or D (dry). (2) Use a simple rain gauge or municipal data; one student rotates the duty. (3) At end of each month, compute \(P_m(\text{rain}) = \frac{\text{No. of R-days}}{\text{Total days in month}}\). (4) Plot the 12 monthly probabilities to reveal seasonal pattern. (5) Compare two consecutive years to test stability.

Assertion–Reason Questions

Assertion (A): Tossing a coin 10 times and getting 7 heads gives empirical \(P(H) = 0.7\).
Reason (R): Empirical probability is computed as (favourable trials)/(total trials).
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
Answer: (a) — A is true (7/10 = 0.7) and R correctly states the empirical formula used.
Assertion (A): Empirical probability is always equal to classical probability.
Reason (R): Both are computed using counts of outcomes.
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
Answer: (d) — A is false: empirical probability varies with the experiment, classical is fixed; they only converge for many trials. R is a true statement on its own.

Frequently Asked Questions

What is empirical probability in Class 9 Maths?

Empirical (or experimental) probability of an event E based on n trials is P(E) = (number of trials in which E occurs) / n. For example, if a coin is tossed 100 times and lands heads 53 times, the empirical probability of heads is 53/100 = 0.53.

How does empirical probability differ from theoretical probability?

Theoretical probability uses logical reasoning about equally likely outcomes (e.g., 1/2 for heads on a fair coin). Empirical probability uses observed frequencies in actual trials. As the number of trials grows large, empirical probability approaches theoretical probability.

What is a sure event and an impossible event in Class 9 probability?

A sure event is one that is certain to occur; its probability is 1 (e.g., rolling a number less than 7 on a standard die). An impossible event cannot occur; its probability is 0 (e.g., rolling a 7 on a standard die).

What is the complementary event rule in Class 9?

If E is an event, then 'not E' is the complementary event. Their probabilities satisfy P(E) + P(not E) = 1. So P(not E) = 1 - P(E). For example, if P(rain) = 0.3, then P(no rain) = 1 - 0.3 = 0.7.

If a die is rolled 60 times and 3 appears 12 times, what is the empirical probability of getting 3?

Empirical probability = (number of times 3 appears) / (total trials) = 12/60 = 1/5 = 0.2. The theoretical probability is 1/6 = 0.1666..., so 0.2 is close to but slightly higher than the theoretical value, as expected for finite samples.

What is the Law of Large Numbers in Class 9 probability?

The Law of Large Numbers, established by Jacob Bernoulli in 1713, states that as the number of trials grows large, the empirical probability of an event approaches its theoretical probability. This is why we expect a fair coin to land heads roughly half the time over many tosses.

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