Expected Outcomes

Expected Outcomes

The expected frequency (or expected number of outcomes) is the number of times an event is predicted to occur if an experiment is repeated many times. It is calculated from the probability of the event and the number of trials.

The Formula

\[ \text{Expected frequency} = P(\text{event}) \times n \]

where \( n \) is the number of trials. This is a theoretical prediction — actual frequencies will vary, but they tend towards the expected frequency as \( n \) increases.

Fairness

An experiment is fair if all outcomes are equally likely. A die is fair if each face has probability 1/6. A spinner is fair if each sector subtends an equal angle. If observed frequencies differ significantly from expected frequencies, the experiment may be biased.

Randomness

A random event is one whose outcome cannot be predicted with certainty. Even in a random experiment, the long-run relative frequency of an outcome tends towards its theoretical probability as the number of trials grows — this is the basis of the expected frequency formula.

Worked Examples

A fair die is rolled 300 times. How many times would you expect to roll a 6? How many times would you expect to roll a number greater than 4?
\[ E(6) = \frac{1}{6}\times300 = 50 \] \[ E(>4) = P(5\text{ or }6)\times300 = \frac{2}{6}\times300 = 100 \]
A spinner has four sections. In 200 spins it lands: Red 62, Blue 48, Green 51, Yellow 39. Is there evidence the spinner is unfair?

Expected frequency for each: \( \frac{1}{4}\times200=50 \). Observed vs expected: Red 62 (high), Yellow 39 (low). Some deviation from 50 is expected due to randomness, but Red being 24% above expected is notable. More trials would be needed to confirm bias.

 Key Takeaways

  • Expected frequency = probability × number of trials.
  • Expected frequency is a prediction — actual results will vary.
  • More trials → relative frequency closer to theoretical probability.
  • A fair experiment has equal probabilities for all outcomes. Evidence of unfairness: observed frequencies differ substantially from expected.