Randomness, fairness, equally likely outcomes

OCR J560 Foundation Probability questions test understanding of fair vs biased and theoretical vs experimental probability. The vocabulary is examined explicitly.

Random and fair

• A trial is random if its outcome cannot be predicted with certainty.
• A device (die, coin, spinner) is fair if every outcome is equally likely.
• A device is biased if some outcomes are more likely than others.

Equally likely outcomes

If there are n equally likely outcomes, the probability of any single outcome = 1/n.

For multiple "successful" outcomes:

• P(event) = (number of successful outcomes) / (total outcomes).

Example: rolling a fair die. P(getting an even number) = 3/6 = 1/2.

Expected outcomes

If you do n trials, the expected number of successful outcomes = n × P(success).

Example: roll a fair die 60 times. Expected number of 6s = 60 × (1/6) = 10.

This is just an estimate — the actual count will fluctuate due to randomness, but in the long run will tend towards 10.

Theoretical vs experimental probability

• Theoretical probability: based on equally likely outcomes (e.g. P(heads) = 1/2 on a fair coin).
• Experimental (relative) frequency: based on observed outcomes.
  • Relative frequency = number of times event occurred / total trials.

For a fair device, experimental probability ≈ theoretical probability, with the agreement improving as n grows.

Detecting bias

Compare experimental and theoretical probabilities. A large discrepancy after many trials suggests bias.

Example: a coin shows 55 heads in 100 flips. Theoretical = 0.5; experimental = 0.55. With only 100 trials, this is well within natural variation. But 5500 heads in 10 000 flips is much stronger evidence of bias.

OCR mark scheme conventions

• "State a reason" → B1 for any valid reason.
• "Calculate the expected number" → M1 for n × p; A1 for the value.
• "Use the relative frequency to estimate the probability" → B1 for the fraction.