Ninth Grade Mathematics: Empirical (Experimental) vs Theoretical Approach
Probability can be approached through two complementary lenses: by observing what actually happens across hundreds of repeated physical trials, or by deducing what must happen based on geometric symmetry and equal likelihood. In this chapter, you will master the distinction between empirical (experimental) probability and theoretical (classical) probability, and explore how the Law of Large Numbers unites them.
The Empirical (Experimental) Approach
Empirical probability is based on actual experimentation, historical records, and observed frequencies. It computes the likelihood of an event based on the ratio of observed successes to the total number of trials conducted:
Empirical (Experimental) Probability Formula:
Number of trials in which the event occurred
P(E) = ----------------------------------------------
Total number of trials conducted
Historically, celebrated statisticians performed thousands of coin tosses to test probability experimentally: Comte de Buffon (18th century): Tossed a coin 4,040 times; obtained 2,048 Heads. Empirical P(H) = 2,048 / 4,040 = 0.5069. J.E. Kerrich (1940s, while interned in WWII): Tossed a coin 10,000 times; obtained 5,067 Heads. Empirical P(H) = 5,067 / 10,000 = 0.5067. Karl Pearson: Tossed a coin 24,000 times; obtained 12,012 Heads. Empirical P(H) = 12,012 / 24,000 = 0.5005.
Empirical probability is indispensable in meteorology (weather forecasting), epidemiology (disease risk), and insurance underwriting, where physical symmetry does not exist.
The Theoretical (Classical) Approach
Theoretical probability (also called classical or a priori probability) does not require performing any physical experiments. Instead, it relies on the assumption that all elementary outcomes of the random experiment are equally likely due to physical symmetry:
Equally Likely Outcomes:
Outcomes are equally likely if there is no physical reason to expect any one
outcome to occur in preference to any of the others.
When flipping a perfectly balanced, symmetrical coin, the physics of the coin gives Heads and Tails identical likelihood. When rolling an unweighted, perfect cube, all six numbered faces are equally likely.
Classical (Theoretical) Probability Formula:
Number of outcomes favorable to Event E n(E)
P(E) = ----------------------------------------- = ------
Total number of possible outcomes in S n(S)
Theoretical Probability of Rolling a 5 on a Die:
Sample Space: S = {1, 2, 3, 4, 5, 6} ===> n(S) = 6
Favorable Event: E = {5} ===> n(E) = 1
P(E) = 1 / 6 approx 0.1667.
The Law of Large Numbers: The Unifying Bridge
How do empirical observation and theoretical deduction connect? Formulated by Swiss mathematician Jakob Bernoulli in 1713, the Law of Large Numbers establishes the connection:
The Law of Large Numbers:
As the number of trials in a random experiment increases toward infinity,
the empirical probability of an event approaches closer and closer to its
theoretical probability.
Convergence of Empirical Probability to Theoretical:
P(Heads)
^
1.0 |
|
0.5 + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Theoretical = 0.5
| /\ /\
| / \ / \/\_/\_/\------------------------------------ Empirical Curve
0.0 +-+----+----+-----+-----+-----+-----+-----+-----+-----+--->
0 10 50 100 500 1000 5000 10000 Number of Tosses (n)
In a small sample of 10 coin tosses, you might observe 7 Heads (empirical probability 0.70). But as you expand to 1,000, 10,000, or 100,000 tosses, random fluctuations average out, and the proportion of Heads converges smoothly to exactly 0.5000.
Chapter Exercises
Exercise 1: A coin is tossed 500 times with the following frequencies: Heads: 265, Tails: 235. Compute the empirical probability for each event, and state their sum.
Exercise 2: Contrast the empirical approach with the theoretical approach by explaining how each method would determine the probability of a drawing pin (thumbtack) landing with its point pointing upward versus a standard six-sided die landing on a 4.
Exercise 3: A tire manufacturing company kept a record of the distance covered before a tire needed to be replaced across 1,000 tires: Less than 4,000 km: 20 tires; 4,000 to 9,000 km: 210 tires; 9,001 to 14,000 km: 325 tires; More than 14,000 km: 445 tires. If a customer buys a tire from this company, find the empirical probability that: (a) it will need to be replaced before covering 4,000 km; (b) it will last more than 9,000 km.
Solutions and Explanations
Solution 1: Total number of trials is N = 500. The empirical probability of getting Heads is P(Heads) = 265 / 500 = 53 / 100 = 0.53. The empirical probability of getting Tails is P(Tails) = 235 / 500 = 47 / 100 = 0.47. The sum of the empirical probabilities of all elementary outcomes is P(Heads) + P(Tails) = 0.53 + 0.47 = 1.00.
Solution 2: A drawing pin has an asymmetrical physical geometry (a broad flat head and a sharp perpendicular point). Because of this asymmetry, the two outcomes—landing point up or point down—are not equally likely, making it impossible to compute the probability theoretically using geometry. The probability of landing point up must be determined empirically by dropping the pin hundreds of times and recording the ratio of point-up landings to total drops. By contrast, a standard die is a symmetrical cube made of uniform density. Its six faces have identical surface areas and centers of gravity, making all six faces equally likely a priori. Therefore, the probability of rolling a 4 can be computed theoretically without rolling the die a single time: P(4) = 1/6.
Solution 3: The total number of tires observed is N = 1,000. For part (a), the number of tires needing replacement before 4,000 km is 20. The empirical probability is P(less than 4,000 km) = 20 / 1,000 = 2 / 100 = 0.02. For part (b), a tire lasting more than 9,000 km includes those lasting 9,001 to 14,000 km (325 tires) and those lasting more than 14,000 km (445 tires). The total favorable count is 325 + 445 = 770 tires. The empirical probability is P(more than 9,000 km) = 770 / 1,000 = 77 / 100 = 0.77.