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Maths · Probability

Experimental probability

Estimate probabilities from experiments using relative frequency, predict how many times an event will happen, and judge whether a coin or dice is fair.

  • 6 key terms
  • All boards
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Warm-up

Answer each one, then check.

  1. 1

    Simplify \(\dfrac{30}{120}\).

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    \(\dfrac{1}{4}\)

  2. 2

    Write \(\dfrac{18}{60}\) as a decimal.

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    0.3

  3. 3

    What is the theoretical probability of heads on a fair coin?

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    \(\dfrac{1}{2}\)

  4. 4

    Work out \(500 \times 0.3\).

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    150

  5. 5

    What is \(\dfrac{1}{6}\) as a decimal, to 2 decimal places?

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    0.17

Learning Objectives

  1. 1Work out relative frequency.
  2. 2Use relative frequency to estimate probability.
  3. 3Predict how many times an event will happen.
  4. 4Decide whether a coin, dice or spinner is fair.

RELATIVE FREQUENCY

Relative frequency is the number of times an event happens divided by the total number of trials.

\(\text{relative frequency} = \dfrac{\text{frequency of the event}}{\text{total number of trials}}\)

Relative Frequency

A spinner is spun 60 times. It lands on red 18 times, blue 30 times and green 12 times. Find the relative frequency of each colour.

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  1. 1 Red \(\dfrac{18}{60} = 0.3\)
  2. 2 Blue \(\dfrac{30}{60} = 0.5\)
  3. 3 Green \(\dfrac{12}{60} = 0.2\)

AnswerRed 0.3, blue 0.5, green 0.2. These add up to 1.

Predicting Outcomes

Use the spinner results to estimate how many times you would expect blue in 500 spins.

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  1. 1 Estimated probability of blue 0.5
  2. 2 Multiply by the number of spins \(0.5 \times 500 = 250\)

AnswerAbout 250 blues.

Is the Dice Fair?

A dice is rolled 120 times and lands on 6 a total of 30 times. Does this suggest the dice is biased?

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  1. 1 Relative frequency of a 6 \(\dfrac{30}{120} = 0.25\)
  2. 2 Expected for a fair dice \(\dfrac{1}{6} = 0.167\), so about 20 sixes
  3. 3 Compare 30 is much bigger than 20

AnswerThe relative frequency 0.25 is well above \(\dfrac{1}{6}\), which suggests the dice may be biased, but more trials would give a more reliable conclusion.

Theoretical or Experimental?

Theoretical probability

  • Worked out from equally likely outcomes.
  • Example: \(P(\text{head}) = \dfrac{1}{2}\).
  • Does not change.

Experimental probability

  • Estimated from the results of trials.
  • Example: 18 heads in 40 tosses gives 0.45.
  • Changes as you do more trials.

How Reliable Is an Estimate?

More trials give better estimates.

  • Small samples

    10 tosses might give 7 heads: 0.7 is a poor estimate of 0.5.

  • Large samples

    1000 tosses will usually give a relative frequency very close to 0.5.

  • Fair

    A fair dice or coin should give relative frequencies close to the theoretical values.

Coin Toss Experiment

Toss a coin 10 times and record the number of heads. Combine class results in groups of 10, 50 and 100 tosses. Work out the relative frequency of heads for each total and comment on how close each is to 0.5.

1. Collect results.

2. Combine.

3. Compare.

A good answer shows: Small groups vary widely (for example 3 or 7 heads out of 10), while the combined class total is very close to 0.5. This shows that relative frequency settles as the number of trials grows.

Can I...?

  1. 1Work out relative frequency.
  2. 2Use it as an estimate of probability.
  3. 3Work out an expected number.
  4. 4Compare experimental and theoretical results.
  5. 5Say whether a dice looks fair.
  6. 6Explain why more trials are better.
  7. 7Show relative frequencies add to 1.
  8. 8Draw a relative frequency graph.

Summary & Exam Focus

  • Relative frequency \(= \dfrac{\text{frequency}}{\text{trials}}\).
  • Expected number \(=\) probability \(\times\) trials.
  • More trials give a more reliable estimate.
  • Compare with the theoretical probability to test fairness.

Exam focus

A spinner is spun 200 times. It lands on red 62 times. Work out an estimate for the probability that it lands on red, and how many reds you expect in 500 spins. (3 marks) (3 marks)

Relative frequency is frequency divided by the number of trials. Multiply by the number of spins for the expected number.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Trial
One repeat of an experiment.
Frequency
How many times an outcome happens.
Relative frequency
Frequency divided by the number of trials.
Theoretical probability
The probability worked out from equally likely outcomes.
Expected number
Probability multiplied by the number of trials.
Fair
Every outcome is equally likely.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Calculator 2 marks

    A spinner is spun 60 times. It lands on red 18 times. Work out the relative frequency of red.

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    Model answer

    \(\dfrac{18}{60} = 0.3\).

    Mark scheme

    • \(\dfrac{18}{60}\) — M1
    • 0.3 — A1
  2. Question 2 Calculator 3 marks

    A spinner is spun 200 times. It lands on red 62 times. Work out an estimate for the number of times it will land on red in 500 spins.

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    Model answer

    The relative frequency is \(\dfrac{62}{200} = 0.31\). The expected number is \(0.31 \times 500 = 155\).

    Mark scheme

    • \(\dfrac{62}{200}\) — M1
    • \(0.31 \times 500\) — M1
    • 155 — A1
  3. Question 3 Calculator 3 marks

    The graph shows the relative frequency of heads as a coin is tossed more and more times. Use it to explain whether the coin is likely to be fair.

    A graph of relative frequency of heads settling close to 0.5 as the number of tosses increases.
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    Model answer

    The relative frequency gets close to 0.5 as the number of tosses increases, so the coin is likely to be fair.

    Mark scheme

    • The relative frequency settles — M1
    • It is near 0.5 — M1
    • Conclusion that the coin is likely to be fair — C1
  4. Question 4 Calculator 3 marks

    A dice is rolled 120 times. It lands on 6 a total of 30 times. Kim says the dice is biased. Is she right? You must show working.

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    Model answer

    The relative frequency of a 6 is \(\dfrac{30}{120} = 0.25\), compared with \(\dfrac{1}{6} \approx 0.17\) for a fair dice. The expected number of sixes is 20. 30 is much higher, so the dice may be biased, but more trials would make the conclusion more reliable.

    Mark scheme

    • \(\dfrac{30}{120} = 0.25\) or expected 20 — M1
    • Comparison with \(\dfrac{1}{6}\) or 20 — M1
    • A conclusion with a reason — C1
  5. Question 5 Calculator 3 marks

    The probability that a bus is late is 0.15. Work out the expected number of late buses in 200 journeys.

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    Model answer

    \(0.15 \times 200 = 30\).

    Mark scheme

    • \(0.15 \times 200\) — M1
    • 30 — A1
    • A sensible unit or statement — B1
  6. Question 6 Explain 2 marks

    Amy tosses a coin 10 times and gets 7 heads. Ben tosses the same coin 1000 times and gets 505 heads. Whose result gives a better estimate of the probability of a head? Give a reason.

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    Model answer

    Ben's result, because he did many more trials, so his relative frequency (0.505) is more reliable.

    Mark scheme

    • Ben — B1
    • More trials give a more reliable estimate — C1

Quick check

  1. A coin lands on heads 12 times in 30 tosses. What is the relative frequency of heads?

    1. A0.12
    2. B0.4
    3. C0.5
    4. D2.5
    Show answerHide answer

    B: 0.4

    \(\dfrac{12}{30} = 0.4\).

  2. The probability of rain is 0.2. How many rainy days do you expect in 100 days?

    1. A2
    2. B12
    3. C20
    4. D200
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    C: 20

    \(0.2 \times 100 = 20\).

  3. Which gives a more reliable estimate of a probability?

    1. AMore trials
    2. BFewer trials
    3. COnly one trial
    4. DA biased guess
    Show answerHide answer

    A: More trials

    More trials give a better estimate.

  4. Relative frequencies for all outcomes add up to...

    1. A0
    2. B0.5
    3. C100
    4. D1
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    D: 1

    They form a probability distribution, so they sum to 1.

  5. A dice is fair. In 600 rolls, about how many sixes do you expect?

    1. A60
    2. B100
    3. C200
    4. D6
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    B: 100

    \(\dfrac{1}{6} \times 600 = 100\).

  6. A coin gives 7 heads in 10 tosses. Is it definitely biased?

    1. AYes, definitely
    2. BYes, because 7 is more than 5
    3. CNo, more trials are needed
    4. DNo, coins are always fair
    Show answerHide answer

    C: No, more trials are needed

    No: with only 10 trials, a result like this can easily happen by chance, so more trials are needed.

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