Probability can be found in two different ways: using theoretical probability and using experimental probability. Comparing these helps judge how reliable results are and whether a theoretical model is appropriate.

 

 

Theoretical Probability

Theoretical probability is calculated using reasoning when all outcomes are equally likely.

 

It is found using:

\( \frac{number\ of\ favourable\ outcomes}{total\ number\ of\ possible\ outcomes} \)

 

Theoretical probability:
• is based on assumptions about fairness
• does not use experimental data
• gives an exact value

 

It represents what is expected to happen in the long term if the experiment is fair.

 

 

Experimental Probability

Experimental probability is estimated from actual results.

 

It is calculated using relative frequency:

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

 

Experimental probability:
• is based on observed data
• can change as more trials are carried out
• gives an estimate, not an exact value

 

Small numbers of trials often give unreliable estimates.

 

 

Comparing the Two Probabilities

To compare probabilities:
• calculate the theoretical probability
• calculate the experimental probability
• compare the two values

 

If the experimental probability is close to the theoretical probability, this suggests:
• the experiment is fair
• the model is reasonable

 

If the values are quite different, this may suggest:
• too few trials were carried out
• the experiment is biased
• outcomes are not equally likely

 

Experimental results are expected to vary, especially with small samples

 

 

Effect of Number of Trials

As the number of trials increases:
• experimental probability usually gets closer to theoretical probability
• random variation has less effect

 

This explains why repeated trials give more reliable results.

 

With very few trials, large differences are normal and expected.

 

 

Interpreting Differences

Differences between experimental and theoretical probability should be explained carefully.

 

Possible reasons include:
• chance variation
• experimental error
• bias in the experiment

 

Differences do not automatically mean the theoretical probability is wrong.

 

 

Drawing Conclusions

When drawing conclusions:
• state both probabilities clearly
• comment on how close they are
• consider the number of trials
• avoid claiming exact agreement

 

Conclusions should be cautious and evidence based.

 

 

Key Points to Remember

Theoretical probability is calculated using equally likely outcomes.
Experimental probability is estimated from results.
Experimental probability can vary from trial to trial.
Larger numbers of trials usually give closer agreement.
Differences should be explained, not ignored.

 

Comparing experimental and theoretical probability helps link mathematical models with real world evidence and supports sensible conclusions about chance and fairness.