Experimental probability and expectation study guide
Probability • Probability
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Key concepts
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Experimental probability and relative frequency
Experimental probability equals the number of times an outcome occurs divided by the total number of trials. The result expresses the observed likelihood of the outcome as a fraction, decimal or percentage on the 0–1 probability scale. Examples include counting heads in coin tosses or red outcomes when spinning a spinner. Relative frequency approximates theoretical probability when trials are unbiased and independent. Differences between experimental and theoretical values occur because of sample variation, measurement error or biased apparatus.
Theoretical probability and equally likely events
Theoretical probability uses known outcomes and equally likely assumptions to calculate exact probabilities. The probability of an event equals favourable outcomes divided by total equally likely outcomes. Fair coins and balanced dice commonly satisfy the equally likely assumption. Incorrect assumptions about fairness or equality of outcomes cause theoretical calculations to misrepresent actual behaviour. The equally likely model applies only when random mechanism and symmetry support identical chances for each outcome.
Expectation and expected outcomes for multiple experiments
Expected value (expectation) multiplies probability by the number of trials to predict average counts of an outcome. For n independent trials with event probability p, the expected number of occurrences equals np. Expected value expresses a long-run average rather than a guaranteed count in a single experiment. Expectation supports planning and comparison of scenarios. Rounding considerations apply when expected counts are not whole numbers: expectation describes an average, not a mandatory integer result for a specific experiment.
Randomness, fairness, and sample size effects
Randomness requires unpredictability of individual outcomes. Fairness requires no systematic bias that favours particular results. When mechanisms are random and fair, empirical unbiased samples tend toward the theoretical distribution as sample size increases (law of large numbers). Smaller samples produce larger relative fluctuations from theoretical probabilities. Persistent deviations from theory suggest bias, dependence between trials, or recording errors rather than natural sample variation.
Recording and analysing frequencies: tables and frequency trees
Frequency tables list outcomes with counts and relative frequencies, enabling quick comparison with theoretical probabilities. Columns often include outcome, count, relative frequency (count ÷ total), and cumulative frequency for ordered data. Frequency trees map sequential experiments or combined events. Branch probabilities multiply along independent branches. Frequency trees with counts replace probabilities with observed counts to display experimental structure and support calculation of combined event frequencies.
Key notes
Important points to keep in mind