Relative & Expected Frequency

Calculating Relative Frequency

• To find the relative frequency, divide the observed frequency of the event by the total number of trials performed:

$\text{Relative Frequency} = \frac{\text{Frequency of Event}}{\text{Total Number of Trials}}$

Calculating Expected Frequency

• To predict the number of occurrences in a set of trials, multiply the probability of the event by the total number of trials ($n$):

$\text{Expected Frequency} = P(\text{Event}) \times n$

• If the theoretical probability is unknown, use the most reliable relative frequency available (the one from the largest sample) as the value for $P(\text{Event})$.

• It is vital to distinguish between what actually happened (Relative Frequency) and what is predicted to happen (Expected Frequency).

• Check the Total: Always ensure you are dividing by the total number of trials, not just the frequency of other outcomes. Summing the frequencies first is a common necessary step.

• Identify the Best Estimate: If an exam provides multiple sets of experimental data, the relative frequency derived from the largest sample size is always the most accurate estimate for probability.

• Sanity Check: Expected frequency does not have to be a whole number. If you calculate that you expect to see $3.5$ heads in $7$ flips, do not round it to $4$ unless the question specifically asks for a whole number of items.

• Bias Detection: Compare the relative frequency to the theoretical probability. If they are significantly different after many trials, the tool (like a die or coin) is likely biased.

• Frequency vs. Relative Frequency: Students often confuse the raw count (frequency) with the ratio (relative frequency). Always check if the answer should be a count or a probability/fraction.

• The Gambler's Fallacy: A common misconception is that if an event has a low relative frequency now, it is 'due' to happen more often soon. In independent trials, the probability remains constant regardless of past results.

• Ignoring Replacement: In sampling scenarios, failing to replace an item changes the probability for the next trial, making the standard relative frequency formula technically incorrect for predicting long-term behavior.