The Mathematics of Maybe: Introduction to Probability · Lesson 4 of 7
Theoretical and Statistical Probability and Long-Run Behaviour
“Theory predicts, data reports, and the long run tries to make them agree.”
• Calculate theoretical probability when outcomes are equally likely. • Distinguish theoretical, experimental and statistical probability. • Understand sampling and representative samples. • Understand the Law of Large Numbers. • Recognise Gambler's Fallacy and independence in repeated fair trials.
Theoretical and Statistical Probability
Experimental probability is based on what actually happens when an experiment is performed many times. We observe the results, count how often a particular event occurs, and use those observations to estimate its probability. The value may change slightly from one set of trials to another because it depends on the outcomes we record.
Theoretical probability is found in a different way. Instead of carrying out the experiment, we study all the possible outcomes and use logical reasoning. When each outcome is equally likely, the probability of an event depends on how many outcomes are favourable to that event compared with the total number of possible outcomes. This allows us to predict the chance of an event before the experiment is actually performed.
Problem
What is the theoretical probability of rolling a 4 on a fair six-sided die?
- 1.There are 6 equally likely outcomes: 1,2,3,4,5,6.
- 2.Only one outcome is favourable: 4.
- 3.P(4)=1/6≈0.167=16.7%.
Problem
One letter is chosen at random from the word PROBABILITY. What is the probability of choosing B?
- 1.The word has 11 letters.
- 2.B appears twice.
- 3.P(B)=2/11≈0.182=18.2%.
When Theoretical Probability Is Appropriate
The formula based on favourable outcomes works only when the possible outcomes can reasonably be treated as equally likely. A fair coin and a fair die are designed for this. A paper cup landing on its bottom, top or side is different: those three outcomes exist, but they are not automatically equally likely.
Statistical Probability from Samples
Sometimes we estimate probability from data collected from a sample. For example, suppose 24 out of 60 surveyed students say mango is their favourite fruit. We might estimate that a randomly selected student has probability 24/60=0.4 of preferring mango.
A smaller group from which data is collected in order to learn about a larger population.
The entire group we want information about.
A larger and more representative sample generally gives us more confidence than a tiny or biased sample. Asking only one class may not represent an entire school as well as asking students from several grades.
Law of Large Numbers
Even for a perfectly fair coin or die, experimental probability may differ from theoretical probability when the number of trials is small. As the number of trials becomes large, the experimental relative frequency tends to move closer to the theoretical probability.
As the number of trials increases, experimental probability tends to get closer to theoretical probability in a fair random experiment.
Problem
A fair die is rolled 12 times and the number 3 appears 3 times. Experimental probability is 3/12=0.25, while theoretical probability is 1/6≈0.167. Is the die necessarily unfair?
- 1.No. A small number of trials can produce noticeable variation.
- 2.Theoretical probability describes the ideal long-run likelihood for a fair die.
- 3.If we repeat the experiment many more times, the relative frequency of 3 would be expected to move closer to 1/6.
- 4.A difference in a small sample is not enough by itself to prove bias.
Gambler's Fallacy
Suppose a fair coin has landed Heads six times in a row. It can feel as though Tails is 'due'. But the coin has no memory. On the next independent toss, the probability of Tails is still 1/2.
The mistaken belief that previous independent random outcomes make the opposite outcome more likely next, simply because one result has occurred repeatedly.
Problem
You roll a fair die three times and get 6 each time. Is rolling another 6 now less likely?
- 1.No.
- 2.Each die roll is an independent event.
- 3.The probability of rolling a 6 remains 1/6 on the next roll.
- 4.The previous rolls do not change the physical faces of the die.
- 5.Thinking that another 6 is now 'due not to happen' is Gambler's Fallacy.
The Law of Large Numbers describes behaviour over many trials. It does not say that short-term results must immediately 'balance out'. A streak can continue even though long-run relative frequencies tend toward theoretical values.
Practice Problems
- A fair die is rolled. Find the theoretical probability of getting an even number.
- Cards showing the letters in the word PEACE are mixed. Find the probability of choosing E and the probability of not choosing E.
- A sample of 50 students contains 18 students who prefer Science Club. Estimate the probability that a randomly selected student from this sample prefers Science Club.
- A fair coin is tossed 8 times and all 8 tosses are Heads. What is the probability of Tails on the ninth toss? Explain.
- A die is rolled 20 times and a 2 appears 6 times. Compare the experimental and theoretical probabilities and explain why they need not match exactly.
- Explain why a sample collected only from one sports team may be a poor sample for estimating the favourite activity of an entire school.
Key Takeaways
• Theoretical probability uses equally likely outcomes. • Experimental probability comes from observed trials. • Statistical estimates often use samples to learn about populations. • Larger, representative samples usually give stronger evidence. • Experimental probability tends toward theoretical probability over many trials. • Independent trials do not remember previous outcomes; believing otherwise leads to Gambler's Fallacy.
Next, we learn to describe all possible outcomes carefully using sample spaces and events.