The Mathematics of Maybe: Introduction to Probability · Lesson 3 of 7
Experimental Probability
“Repeat an experiment enough times and randomness starts revealing its habits.”
• Understand outcome, sample space and trial in simple experiments. • Calculate experimental probability. • Interpret relative frequency. • Use observed data to estimate likelihood. • Understand why experimental results can vary from one set of trials to another.
Introduction
Sometimes, instead of predicting how likely an event is based only on reasoning, we can find out by actually performing the experiment several times and recording the results. Each repetition gives us evidence about how often a particular outcome occurs.
Experimental probability is the probability estimated from these observed results. It is found by comparing the number of times an event occurs with the total number of trials. The more times an experiment is repeated, the more reliable this estimate usually becomes.
A result produced by a random experiment.
The set of all possible outcomes of a random experiment.
Problem
What are the possible outcomes when tossing a coin and when rolling a standard die?
- 1.Coin: S={H,T}.
- 2.Die: S={1,2,3,4,5,6}.
- 3.The sample space lists every possible outcome exactly once.
The fraction or decimal obtained by dividing the number of times an event occurs by the total number of observations.
Problem
A die is rolled 40 times and the number 5 appears 7 times. Find the experimental probability of rolling a 5.
- 1.Event count=7.
- 2.Total trials=40.
- 3.Experimental probability=7/40.
- 4.7/40=0.175=17.5%.
Daily-Life Example: A Paper Cup
Suppose you toss a paper cup repeatedly. It may land upright on its bottom, upside down on its top, or on its side. These outcomes are not obviously equally likely, so simply counting outcomes is not enough. Repeating the toss many times and recording the results gives useful experimental probabilities.
Experimental Probability Can Change
If you toss a coin 10 times, you might get 7 heads. Another student might get 4 heads. Experimental probability depends on the data actually observed, so small sets of trials may give noticeably different values.
Problem
A classroom spinner is used 20 times and lands on blue 9 times. What can we say?
- 1.The experimental probability of blue is 9/20=0.45.
- 2.This describes what happened in these 20 trials.
- 3.It does not prove that the true long-run probability is exactly 0.45.
- 4.More trials would provide more evidence.
Practice Problems
- A coin is tossed 30 times and shows Heads 17 times. Find the experimental probability of Heads and Tails.
- A paper cup is tossed 60 times and lands on its side 31 times. Find the relative frequency of landing on its side.
- A spinner lands on red 12 times in 50 spins. Calculate the experimental probability of red.
- Two students repeat the same coin experiment 20 times but get different experimental probabilities. Explain why this is possible.
- Design a simple random experiment you can perform at home and state what data you would record to estimate an experimental probability.
Key Takeaways
• Experimental probability comes from observed data. • Relative frequency and experimental probability use the same ratio. • Small experiments can give different results from one another. • Repeating an experiment gives more evidence about long-run behaviour. • Experimental probability describes observed frequency, not a guaranteed next result.
Next, we compare experimental probability with theoretical probability and statistical estimates, then study the Law of Large Numbers and Gambler's Fallacy.