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Chapter overview

Maths

Exam board

AQA

Number

Algebra

Graphs

Ratio proportion and rates of change

Shapes and area

Angles and geometry

Trigonometry

Probability

Statistics

Maths

Probability: Expected and relative frequency

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Probability: Expected and relative frequency

​​In a nutshell

Probabilities are usually theoretical value that indicate the chance of an event occurring, however probabilities can be estimated from relative frequency. The more times an experiment is run, the better an estimate of the true probabilities the relative frequencies are.



Expected frequency

​​Definition

The expected frequency of an outcome is the number of times the outcome is expected to occur. You can work it out using the following formula:


expected frequency=theoretical probability×number of trials\text{expected frequency} = \text{theoretical probability} \times \text{number of trials}


Example 1

Ian is testing whether or not a coin is fair by flipping the coin 100100​ times and recording the results. How many times should he expect the coin to land heads?


The probability of heads is 12\dfrac 1 2. Use the formula to find the expected frequency.

Expected frequency = 12×100=50‾\dfrac{1}{2} \times 100 = \underline{50}.



Relative frequency

Definition

The relative frequency of an outcome is the proportion of times the outcome has occurred out of all the outcomes. If an experiment is run for enough trials, it can be used as an estimate of the true probability of an outcome. You can work it out using the following formula:


​relative frequency=frequency of outcomenumber of trials\text{relative frequency} = \dfrac{\text{frequency of outcome}}{\text{number of trials}}


Example 2

Ian finds that from the 100100​ flips, the coin lands heads 3737 times. What is the relative frequency of heads?


relative frequency = 37100=0.37‾\dfrac{37}{100} = \underline{0.37}.



Typical random experiments


Coin flip

Die roll

Matchsticks

Maths; Probability; KS4 Year 10; Probability: Expected and relative frequency
Maths; Probability; KS4 Year 10; Probability: Expected and relative frequency
Maths; Probability; KS4 Year 10; Probability: Expected and relative frequency
Probability of:
Heads =12=\dfrac{1}{2}, tails =12=\dfrac{1}{2}​\dfrac{​
Probability of:
 P(1)=16P(1) =\dfrac{1}{6}​​
Probability of:
Short =13=\dfrac{1}{3}, long =23=\dfrac{2}{3}​​




Frequently Asked Questions (FAQ)

FAQs

  • Question: How does the sample size affect the relative frequency?

    Answer: As the sample size increases, the relative frequency becomes a more accurate estimate of the true probability of an outcome.

  • Question: What are some examples of probability experiments?

    Answer: Throwing a die, tossing a coin and rotating a spinner are all examples of probability experiments. A trial produces one and only one outcome from all the possible outcomes.

  • Question: What is experimental probability?

    Answer: Experimental probability is the same as relative frequency, which is the proportion of times an outcome occurs in an experiment out of the total number of trials.

Theory

Exercises

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