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Probabillity and Statistics – Experimental probability

📘 Notes

Statistical Probability

Statistical probability is based on what actually happens in an experiment or in real life. It is sometimes called experimental probability.

For example, instead of using a perfect maths model, we use results from trials, records, or data to estimate the chance of an event happening.

Key facts and formulae

1. Experimental probability formula

Experimental probability is found using data:

\[ \text{Experimental probability}=\frac{\text{number of times the event happened}}{\text{total number of trials}}. \]

2. Using data to estimate probability

If an event has happened many times in the past, we can use those results to estimate the probability that it will happen again.

The estimate becomes more reliable when the number of trials is larger.

3. Statistical probability and theoretical probability

Type Based on
Theoretical probability All equally likely outcomes
Statistical probability Results from experiments or data

4. Writing answers

Statistical probability can also be written as a fraction, decimal, or percentage.

\[ \frac{8}{20}=0.4=40\% \]

Always read the question carefully to see which form is needed.

Worked examples

Example 1: Late for school

A student was late for school on \(4\) days out of \(20\) school days. Estimate the probability that the student will be late tomorrow.

Solution

\[ \text{Experimental probability}=\frac{4}{20}=\frac{1}{5}. \]
The estimated probability is \[ \frac{1}{5}. \]

Example 2: Rainy days

In the last \(30\) days, it rained on \(12\) days. Estimate the probability that it will rain tomorrow.

Solution

\[ \text{Experimental probability}=\frac{12}{30}=\frac{2}{5}. \]
The estimated probability is \[ \frac{2}{5}. \]

Example 3: Traffic lights

Over \(50\) journeys, a driver had to stop at a traffic light \(32\) times. Estimate the probability that the driver will have to stop next time.

Solution

\[ \text{Experimental probability}=\frac{32}{50}=\frac{16}{25}. \]
The estimated probability is \[ \frac{16}{25}. \]

Example 4: Not snowing

In the last \(40\) winter days, it snowed on \(6\) days. Estimate the probability that it will not snow tomorrow.

Solution

Probability of snow:

\[ \frac{6}{40}=\frac{3}{20}. \]

Probability of not snowing:

\[ 1-\frac{3}{20}=\frac{17}{20}. \]
The estimated probability is \[ \frac{17}{20}. \]

Common mistakes and exam tips

Common mistakes

  • Using the wrong total number of trials.
  • Mixing up the number of times the event happened with the number of times it did not happen.
  • Forgetting to use the word estimate.
  • Not simplifying the final fraction.

Exam tips

  • Look for the event count and the total number of trials.
  • Use past data exactly as given in the question.
  • If the question asks for “not”, use the complement.
  • Large amounts of data usually give a better estimate.

Summary

\[ \text{Experimental probability}=\frac{\text{number of times event happened}}{\text{total number of trials}} \] \[ P(\text{not }A)=1-P(A) \]

Statistical probability uses real results and data to estimate how likely something is to happen.