Sampling and convergence Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Sampling and convergence questions. See exactly how to solve problems on relative frequency, experimental probability, simplifying fractions, expected frequency.

relative frequencyexperimental probabilitysimplifying fractionsexpected frequencyprobability x trialsobserved frequency
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
A drawing pin is dropped 200200 times. It lands point up 7474 times. Work out the relative frequency of landing point up. Give your answer as a decimal.

Worked solution

  1. Write down the rule for relative frequency

    relative frequency=number of successesnumber of trials\text{relative frequency} = \frac{\text{number of successes}}{\text{number of trials}}

    Relative frequency (experimental probability) is worked out from the results of an experiment: how many times the event happened, divided by how many times the experiment was carried out.

  2. Substitute the results of the experiment

    relative frequency=74200\text{relative frequency} = \frac{74}{200}

    The event landing point up happened 7474 times in 200200 trials, so the relative frequency is 74200\frac{74}{200}.

  3. Work out the answer

    74200=0.37\frac{74}{200} = 0.37

    Dividing 7474 by 200200 gives 0.370.37. This is an ESTIMATE of the probability of landing point up, not the probability itself.

Answer
0.370.37
Question 2
1 markeasy
A biased coin is thrown 5050 times. It lands on heads 3232 times. Work out the relative frequency of landing on heads. Give your answer as a decimal.

Worked solution

  1. Write down the rule for relative frequency

    relative frequency=number of successesnumber of trials\text{relative frequency} = \frac{\text{number of successes}}{\text{number of trials}}

    Relative frequency (experimental probability) is worked out from the results of an experiment: how many times the event happened, divided by how many times the experiment was carried out.

  2. Substitute the results of the experiment

    relative frequency=3250\text{relative frequency} = \frac{32}{50}

    The event landing on heads happened 3232 times in 5050 trials, so the relative frequency is 3250\frac{32}{50}.

  3. Work out the answer

    3250=0.64\frac{32}{50} = 0.64

    Dividing 3232 by 5050 gives 0.640.64. This is an ESTIMATE of the probability of landing on heads, not the probability itself.

Answer
0.640.64
Question 3
2 markseasy
A biased spinner is spun 8080 times. It lands on red 2828 times. Work out the relative frequency of landing on red. Give your answer as a fraction in its simplest form.

Worked solution

  1. Write down the rule for relative frequency

    relative frequency=number of successesnumber of trials\text{relative frequency} = \frac{\text{number of successes}}{\text{number of trials}}

    Relative frequency (experimental probability) is worked out from the results of an experiment: how many times the event happened, divided by how many times the experiment was carried out.

  2. Substitute the results of the experiment

    relative frequency=2880\text{relative frequency} = \frac{28}{80}

    The event landing on red happened 2828 times in 8080 trials, so the relative frequency is 2880\frac{28}{80}.

  3. Work out the answer

    2880=720\frac{28}{80} = \frac{7}{20}

    Dividing the numerator and the denominator by 44 gives 720\frac{7}{20} in its simplest form. This is an ESTIMATE of the probability of landing on red, not the probability itself.

Answer
720\frac{7}{20}
Question 4
2 markseasy
A drawing pin is dropped 150150 times. It lands point down 9090 times. Work out the relative frequency of landing point down. Give your answer as a fraction in its simplest form.

Worked solution

  1. Write down the rule for relative frequency

    relative frequency=number of successesnumber of trials\text{relative frequency} = \frac{\text{number of successes}}{\text{number of trials}}

    Relative frequency (experimental probability) is worked out from the results of an experiment: how many times the event happened, divided by how many times the experiment was carried out.

  2. Substitute the results of the experiment

    relative frequency=90150\text{relative frequency} = \frac{90}{150}

    The event landing point down happened 9090 times in 150150 trials, so the relative frequency is 90150\frac{90}{150}.

  3. Work out the answer

    90150=35\frac{90}{150} = \frac{3}{5}

    Dividing the numerator and the denominator by 3030 gives 35\frac{3}{5} in its simplest form. This is an ESTIMATE of the probability of landing point down, not the probability itself.

Answer
35\frac{3}{5}
Question 5
1 markeasy
A biased dice is rolled 500500 times. It lands on a six 9595 times. Work out the relative frequency of landing on a six. Give your answer as a decimal.

Worked solution

  1. Write down the rule for relative frequency

    relative frequency=number of successesnumber of trials\text{relative frequency} = \frac{\text{number of successes}}{\text{number of trials}}

    Relative frequency (experimental probability) is worked out from the results of an experiment: how many times the event happened, divided by how many times the experiment was carried out.

  2. Substitute the results of the experiment

    relative frequency=95500\text{relative frequency} = \frac{95}{500}

    The event landing on a six happened 9595 times in 500500 trials, so the relative frequency is 95500\frac{95}{500}.

  3. Work out the answer

    95500=0.19\frac{95}{500} = 0.19

    Dividing 9595 by 500500 gives 0.190.19. This is an ESTIMATE of the probability of landing on a six, not the probability itself.

Answer
0.190.19

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