When outcomes are uncertain or not equally likely, probability can be estimated from experimental evidence. This is done by looking at how often an event has occurred and calculating its relative frequency.

 

 

Experimental Probability

Experimental probability is based on the results of an experiment or trial.

 

It uses real data rather than theoretical reasoning.

 

The probability of an event is estimated by comparing:
• how many times the event occurred
• how many trials were carried out

 

This estimate becomes more reliable as the number of trials increases.

 

 

Relative Frequency

Relative frequency describes how often an event occurs compared to the total number of trials.

 

It is calculated as:

\( \frac{number\ of\ times\ the\ event\ occurs}{total\ number\ of\ trials} \)

 

This fraction represents the estimated probability of the event.

 

Relative frequency is always between \( 0 \) and \( 1 \).

 

 

Link Between Probability and Evidence

Each time an experiment is repeated, more evidence is collected.

 

As the number of trials increases:
• the relative frequency often settles down
• the estimate becomes more stable

 

This means the estimated probability becomes closer to the true probability.

 

This idea is sometimes called the long run behaviour of probability

 

For example, if an event has occurred 30 times out of 100 trials, the estimated probability is:

\( \frac{30}{100} \)

 

This estimate is based entirely on experimental evidence.

 

 

Reliability of Estimates

Estimated probabilities depend on:
• the number of trials
• how fairly the experiment is carried out
• whether conditions stay the same

 

Small numbers of trials can give misleading estimates.

 

Larger numbers of trials usually give more reliable results.

 

One result does not determine probability

 

 

Comparing Experimental and Theoretical Probability

Experimental probability:
• is based on data
• may change as more trials are done
• is useful when outcomes are not equally likely

 

Theoretical probability:
• is based on reasoning
• assumes equally likely outcomes

 

Experimental results can be used to check whether a theoretical model is reasonable.

 

 

Common Errors to Avoid

Common mistakes include:
• assuming small samples give accurate probabilities
• forgetting to divide by the total number of trials
• treating experimental estimates as exact values

 

Always describe results as estimates.

 

 

Key Points to Remember

Experimental probability is based on observed results.
Relative frequency is the proportion of times an event occurs.
Estimated probability is found by dividing occurrences by trials.
Estimates become more reliable with more trials.
Experimental probability links chance to real evidence.

 

Estimating probability using relative frequency allows uncertainty to be measured using real data and supports informed conclusions based on experimental evidence.