Relative & Expected Frequency | Mathematics for GCSE/IGCSE - Year 11 PDF Download

Relative Frequency

What is relative frequency?

• Relative frequency is used to estimate probabilities from experimental data
  • For a certain number of trials, the probability of ‘success’ is given as:
    
  • For Example, If you flip an unfair coin 50 times and it lands on heads 20 times then you would use relative frequency to estimate the probability of the coin landing on heads as 20/50
• Increased trials enhance the precision of relative frequency.
  • When estimating probability based on relative frequencies, prioritize the one derived from the highest number of trials.

When will I be asked to use relative frequency?

• Relative frequency serves as a substitute for theoretical probabilities when the latter are unavailable or impractical to compute.
• Relative frequency can also serve as a means to assess the fairness or bias of a situation.
  • For instance, in the case of a fair coin, the expected probability of it landing on heads is 0.5.
    • A relative frequency close to 0.5 indicates fairness.
    • Conversely, a relative frequency significantly different from 0.5 suggests bias or unfairness.

What else do I need to know about relative frequency?

• Relative frequency serves as an approximation for probability.
  • With a large number of trials, the estimate tends to converge towards the actual probability.
• Relative frequency operates under the assumption of equal chances of success on each trial.
  • This implies that trials are independent.
  • In situations involving selecting items from a bag (like buttons, balls, or marbles), replacement is necessary to maintain the conditions for using relative frequency.

Expected Frequency

What is expected frequency?

• Expected frequency refers to the number of times you would expect a particular outcome to occur when repeating a trial numerous times.
• The theoretical probability of that outcome will need to be known or an estimate of it, from relative frequency.

How do I find expected frequency?

• When the probability of a specific outcome is p and there are n trials:
  • The expected number of occurrences of that outcome from the n trials is np.
  • This is achieved by multiplying the number of trials by the probability of the particular outcome.
• It's important to understand that this doesn't guarantee precisely np occurrences.
  • However, if the experiment (of n trials) were repeated numerous times, we would anticipate the average number of occurrences to approach np.