Comparing Experimental and Theoretical Probability - Seventh Grade Mathematics
The Law of Large Numbers connects experimental trials to theoretical expectations, demonstrating that relative frequencies stabilize and converge toward true probabilities as trials multiply.
Short-Term Volatility versus Long-Term Stability
When you flip a fair coin ten times, theoretical probability states that heads has a fifty percent probability. Does that guarantee you will observe exactly five heads and five tails? Absolutely not. You might observe seven heads (70%) or three heads (30%) simply through random short-term variation.
THE PARADOX OF SHORT-TERM CHANCE
Short-term trials (n = 10): Highly volatile; wide divergence from theory.
Long-term trials (n = 10,000): Highly stable; converges tightly to theory.
In the short run, random chance produces noticeable fluctuations. In the long run, random fluctuations average out, revealing the underlying mathematical balance.
The Law of Large Numbers
The Law of Large Numbers is a foundational theorem of probability theory: as the number of experimental trials increases, the experimental relative frequency of an event approaches closer and closer to its theoretical probability.
THE LAW OF LARGE NUMBERS
As Total Trials n ------> Infinity:
Experimental Probability ------> Theoretical Probability
Consider simulating a fair coin flipped thousands of times, tracking the cumulative percentage of heads:
Cumulative Percent of Heads:
100% | *
| 75% | *
| 50% |-------*----*----*--*--*-*--*--- [ Theoretical Line: 50% ]
|
25% |
+----+----+----+----+----+----+----+> Total Flips
10 50 100 500 1000 5000 10000
During the first ten to fifty flips, the relative frequency oscillates wildly. But by five hundred, one thousand, and ten thousand flips, the cumulative percentage hugs the theoretical fifty-percent line with microscopic deviation.
Comparing Experimental and Theoretical Discrepancies
When analyzing the difference between experimental results and theoretical expectations, calculate the absolute difference:
PROBABILITY DISCREPANCY
Discrepancy = |Experimental Probability - Theoretical Probability|
Consider rolling a standard six-sided die sixty times. Theoretical probability predicts each number should appear one-sixth of the time: sixty times one-sixth equals ten times.
+---------------+-------------------+-------------------+--------------------+
| Outcome Face | Theoretical Count | Observed Count | Experimental Prob. |
+---------------+-------------------+-------------------+--------------------+
| 1 | 10 | 8 | 8 / 60 = 13.3% |
| 2 | 10 | 11 | 11 / 60 = 18.3% |
| 3 | 10 | 9 | 9 / 60 = 15.0% |
| 4 | 10 | 13 | 13 / 60 = 21.7% |
| 5 | 10 | 10 | 10 / 60 = 16.7% |
| 6 | 10 | 9 | 9 / 60 = 15.0% |
+---------------+-------------------+-------------------+--------------------+
Although no single face appeared exactly ten times except for five, every observed count hovered near ten. The discrepancies are normal consequences of random variation.
Detecting Unfair or Biased Experimental Tools
The Law of Large Numbers provides an investigative tool for detecting fraud or physical bias in gaming equipment. If a coin or die is fair, its experimental relative frequency MUST converge toward its theoretical probability over thousands of trials.
Testing a Suspicious Die over 6,000 Rolls:
Theoretical expectation for rolling a 6: (1/6) * 6000 = 1,000 times (16.7%)
Observed rolls of 6: 2,400 times (40.0%!)
Discrepancy: |40.0% - 16.7%| = 23.3% over 6,000 trials!
Over six thousand rolls, random fluctuation cannot account for a twenty-three percent discrepancy. We can conclude with mathematical certainty that the die is physically weighted or biased toward the number six.
Exercises
Work through each problem carefully in your notebook before reading the provided solutions.
Exercise 1. A student spins a four-color spinner (with equal red, blue, yellow, and green sectors) twenty times and records: Red 7, Blue 6, Yellow 4, Green 3. Compare the theoretical probability of landing on red with its experimental probability.
Exercise 2. A computer program simulates flipping a fair coin. After twenty flips, the relative frequency of heads is seventy percent. After two thousand flips, the relative frequency of heads is fifty point four percent. Explain how this demonstrates the Law of Large Numbers.
Exercise 3. A bag contains an unknown mixture of marbles. In five hundred random draws with replacement, three hundred twenty draws produce a blue marble. What is your best estimate of the theoretical probability of drawing a blue marble?
Exercise 4. A game manufacturer tests a new six-sided die by rolling it twelve times. The number five appears four times. Can the manufacturer conclude from this test that the die is loaded? Explain why or why not.
Solutions
Solution to Exercise 1. For a four-color spinner with equal sectors, the theoretical probability of landing on red is one-fourth, which equals zero point twenty-five (25%). In the experiment of twenty spins, red appeared seven times. The experimental probability is seven-twentieths, which equals zero point thirty-five (35%). The experimental probability exceeded the theoretical probability by ten percentage points (35% - 25% = 10%) due to normal short-term random variation.
Solution to Exercise 2. In the short run of twenty flips, random chance produced a wide deviation: seventy percent heads compared to the theoretical expectation of fifty percent (a twenty-percentage-point error). However, when the number of trials increased one hundred-fold to two thousand flips, the cumulative relative frequency settled to fifty point four percent, differing from the theoretical fifty percent by less than half a percentage point (0.4%). This illustrates the Law of Large Numbers: increasing trials dampens short-term random volatility and forces the experimental rate to converge toward theoretical probability.
Solution to Exercise 3. According to the Law of Large Numbers, a large sample of five hundred trials provides a highly stable and reliable approximation of the true theoretical probability. Calculate the experimental relative frequency: three hundred twenty divided by five hundred. Simplify by dividing: three hundred twenty divided by five hundred equals sixty-four hundredths (0.64), or sixty-four percent (64%). The best estimate of the theoretical probability of drawing a blue marble is sixty-four percent (or 16/25).
Solution to Exercise 4. No, the manufacturer cannot conclude that the die is loaded based on only twelve rolls. Twelve trials is an extremely small sample size subject to severe short-term volatility. In twelve rolls, seeing an outcome occur four times instead of the expected two times happens routinely by pure random chance. To make a scientifically valid assessment about whether the die is loaded, the manufacturer must conduct a much larger experiment consisting of hundreds or thousands of rolls.