Experimental Probability and Relative Frequency - Seventh Grade Mathematics
Experimental probability measures the empirical likelihood of an event based on observed frequency data collected through repeated experimental trials.
The Nature of Experimental Probability
While theoretical probability predicts what should happen based on mathematical symmetry, experimental probability measures what actually happens when an experiment is executed. Experimental probability is often called relative frequency.
THE EXPERIMENTAL PROBABILITY FORMULA
P_exp(E) = ( Number of times event E occurred ) / ( Total number of trials )
= Frequency / Total Trials
Experimental probability is grounded in observed data. It is the primary form of probability used in fields where theoretical symmetry does not exist: weather forecasting, sports statistics, medical efficacy trials, and financial risk modeling.
+-----------------------+-----------------------------+------------------------+
| Feature | Theoretical Probability | Experimental Prob. |
+-----------------------+-----------------------------+------------------------+
| Basis of Calculation | Mathematical symmetry & rules| Real-world trial data |
| Formula | Favorable / Total Possible | Frequency / Trials |
| Prior Execution Needed| No trials needed | Must run trials first |
| Primary Applications | Games of chance, geometry | Science, sports, risk |
+-----------------------+-----------------------------+------------------------+
Recording Trials with Frequency Tables
To determine experimental probability, researchers record the outcomes of repeated trials in structured frequency distribution tables.
Experiment: A student spins an unmarked mystery spinner 50 times.
+---------------+-----------------------+-----------------------+
| Color Outcome | Tally Frequency | Relative Frequency |
+---------------+-----------------------+-----------------------+
| Red | 14 | 14 / 50 = 0.28 = 28% |
| Blue | 21 | 21 / 50 = 0.42 = 42% |
| Green | 15 | 15 / 50 = 0.30 = 30% |
+---------------+-----------------------+-----------------------+
Total Trials: 50. Sum of Relative Frequencies: 0.28 + 0.42 + 0.30 = 1.00 (100%)
From this experimental record, the experimental probability of the spinner landing on blue is twenty-one fiftieths, or forty-two percent.
Real-World Applications of Relative Frequency
Many of the most important metrics in modern society represent experimental probabilities:
Batting Average in Baseball:
A baseball player gets 48 hits in 160 at-bats.
P_exp(Hit) = 48 / 160 = 0.300 (A ".300 batting average")
The player has a 30% empirical chance of getting a hit.
Quality Control in Manufacturing:
A factory inspects 500 circuit boards and finds 4 defective.
P_exp(Defect) = 4 / 500 = 0.008 = 0.8% defect rate.
In both examples, no mathematical formula could deduce the probability beforehand; the probability can only be discovered by testing and measuring real-world occurrences.
Exercises
Work through each problem carefully in your notebook before reading the provided solutions.
Exercise 1. A student tosses a coin eighty times and records forty-eight heads and thirty-two tails. Calculate the experimental probability of tossing heads as a fraction in simplest terms, as a decimal, and as a percent.
Exercise 2. A basketball player attempts two hundred free throws during basketball practice and successfully makes one hundred fifty-six of them. What is the player's experimental probability of making a free throw?
Exercise 3. A quality control technician tests six hundred smartphone screens and detects nine cracked screens. Based on this experiment, how many cracked screens would you predict in a production run of twenty thousand smartphones?
Exercise 4. A soccer goalkeeper faces forty penalty kicks during a season and saves twelve of them. Calculate the experimental probability that the goalkeeper does NOT save a penalty kick.
Solutions
Solution to Exercise 1. The total number of experimental trials is eighty. The event of tossing heads occurred forty-eight times. Calculate the relative frequency: forty-eight divided by eighty. Simplify the fraction by dividing numerator and denominator by sixteen: forty-eight divided by sixteen is three, and eighty divided by sixteen is five. In fractional form: three-fifths (3/5). In decimal form: three divided by five equals zero point six (0.6). In percent form: sixty percent (60%). The experimental probability of tossing heads is three-fifths, or 60%.
Solution to Exercise 2. The number of successful free throws is one hundred fifty-six out of two hundred attempts. Calculate the experimental probability: one hundred fifty-six divided by two hundred. Divide numerator and denominator by two to express over one hundred: seventy-eight over one hundred. The player's experimental probability of making a free throw is seventy-eight hundredths (0.78), or seventy-eight percent (78%).
Solution to Exercise 3. First determine the experimental probability of a defective screen: nine divided by six hundred. Divide numerator and denominator by three: three two-hundredths (0.015 or 1.5% defect rate). To predict the number of defective screens in a production run of twenty thousand smartphones, multiply the total production by the experimental probability: twenty thousand times zero point zero one five. Twenty thousand times zero point zero one five equals three hundred. You would predict approximately three hundred cracked screens in the production run.
Solution to Exercise 4. The goalkeeper saved twelve out of forty penalty kicks. The number of penalty kicks that were not saved is forty minus twelve, which equals twenty-eight kicks. Calculate the experimental probability of not saving: twenty-eight divided by forty. Divide numerator and denominator by four: seven-tenths (7/10). In decimal form: zero point seven (0.7). In percent form: seventy percent (70%). The experimental probability that the goalkeeper does not save a penalty kick is seventy percent.