Short on time? This page is the recap. For explanations and worked examples, use the full study guide.
Recap
- A random process has results decided by chance. One repetition is a trial.
- An outcome is the result of one trial. An event is a collection of outcomes.
- The probability of an event is its relative frequency over a very large number of trials.
- A relative frequency from real or simulated data estimates the true probability.
- A simulation uses random digits or random integers to imitate the random process.
Key relationships
| Idea | What to remember |
|---|---|
| Estimated probability | number of trials with the event ÷ total number of trials |
| Law of large numbers | with independent trials, the relative frequency settles towards one value (the probability) as trials increase |
| Probability 0.1, 0.2, … | single digits: 1, 2, … of the digits 0 to 9 |
| Probability 0.35, 0.07, … | pairs of digits: 35, 7, … of the pairs 00 to 99 |
| Outcomes that do not fit 10 digits | assign some digits and ignore the rest (e.g. 1 to 3 for three equally likely outcomes) |
| Without replacement | skip a value that repeats within a trial |
| With replacement / independent parts | repeats are allowed |
Simulation steps
- State the event and the probability of each basic outcome.
- Assign random values to outcomes in the right proportions; say which values to ignore and whether to skip repeats.
- Describe one trial and what you record.
- Do many trials; record the count and the total.
- Estimate the probability and interpret it in context.
Mistakes to avoid
- “0.2 means exactly 1 in 5.” Only in the long run, and only approximately.
- Believing an outcome is “due” after a run without it. Independent trials have no memory.
- Assigning the wrong number of digits (0 to 2 is three digits, so probability 0.3).
- Not skipping repeats when the same item cannot be chosen twice in a trial.
- Calling a simulated estimate exact. Another run gives a different value.
- Too few trials, or giving a count with no total.
- No context in the conclusion.
Quick self-check
- Assign pairs of random digits to simulate an event with probability 0.45. (Pairs 00 to 44 = event, 45 to 99 = not the event: 45 of 100 pairs)
- In 400 simulated trials an event happened 92 times. Estimate its probability. (92 ÷ 400 = 0.23)
- A fair spinner has landed on red 5 times in a row. Is “not red” more likely next time? (No. Spins are independent, so the probabilities for the next spin are unchanged.)
Next: practice questions.