Cambridge Lower Secondary CheckpointStage 7

Statistics and Probability (7Sp)

Mathematics Stage 7 Chapter Notes

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Statistics and Probability - Probability
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In this chapter

  1. 1Understanding Sampling
  2. 2Mean, Median, Mode, and Range
  3. 3Organising Data: Frequency Tables
  4. 4The Probability Scale
  5. 5Calculating Theoretical Probability
  6. 6Experimental Probability & Relative Frequency

1. Understanding Sampling

In statistics, a 'population' is the entire group we are interested in (e.g., all employees in a company). It's often impractical to collect data from everyone, so we take a 'sample', which is a smaller, manageable subgroup. The goal is to choose a 'representative sample' that accurately reflects the characteristics of the whole population. A biased sample, such as only asking your friends, will not give reliable conclusions because it doesn't represent the variety within the larger group.

Key term

Representative Sample: A sample whose characteristics accurately reflect those of the larger population from which it is drawn.

Examiner insight

Examiners look for a clear explanation of *why* a sample is biased, often by linking the sampling method to a specific characteristic that makes the group different from the general population.

Common pitfall

Assuming that a 'random' sample is always 'representative'. A small random sample can still be unrepresentative by pure chance, which is why larger sample sizes are generally better.

Worked example 12 marks

A school has 150 students in Year 7. A teacher wants to investigate their opinions on school lunches. She decides to give a questionnaire to a sample of 30 students. Explain why it would not be a good idea to choose 30 students from the school's football team.

  1. 1
    1. Identify the population and the sample: The population is all 150 Year 7 students. The sample is the 30 students from the football team.
  2. 2
    1. Identify the potential for bias: The football team is a specific group with a shared interest. Their opinions might not be typical of all students. For example, they may have different dietary needs or preferences due to their sport.
  3. 3
    1. State the conclusion: This is a biased sample because it is not representative of the entire Year 7 population. The results of the questionnaire would likely not reflect the views of all 150 students.

Recap

  • A population is the entire group of interest.
  • A sample is a smaller subset of the population.
  • A biased sample does not accurately represent the population.
  • A representative sample should reflect the characteristics of the whole population.
  • Convenience sampling, like asking friends, often leads to bias.

Quick check

  1. To find out the favourite TV show of students in a school, a researcher surveys the first 20 students arriving at the school gate. Is this sample likely to be representative? Why?2 marks

2. Mean, Median, Mode, and Range

These are key measures used to summarize a set of data. The 'mean', 'median', and 'mode' are types of averages, telling us about the center of the data. The 'range' is a measure of spread, telling us how varied the data is.

  • The Mode is the most frequent value.
  • The Median is the middle value once the data is in order.
  • The Mean is calculated by adding all values and dividing by the count of values.
  • The Range is the difference between the highest and lowest values.

Mean = (Sum of all values) ÷ (Number of values)

Range = Highest value – Lowest value

Key term

Median: The middle value in a dataset when the values are arranged in numerical order.

Examiner insight

Marks are often awarded for showing the correct formula for the mean, even if a calculation error is made. Always write down your method, like 'Sum of values / Number of values'.

Common pitfall

Forgetting to order the data from smallest to largest before finding the median. This is the most common error and leads to an incorrect answer.

Worked example 15 marks

A teacher records the test scores of 8 students: 5, 8, 9, 4, 5, 10, 6, 8. Find the mode, median, mean, and range of the scores.

  1. 1
    1. Order the data: 4, 5, 5, 6, 8, 8, 9, 10.
  2. 2
    1. Find the Mode (most frequent): The values 5 and 8 both appear twice. So the data is bimodal with modes 5 and 8.
  3. 3
    1. Find the Range (highest - lowest): Range = 10 - 4 = 6.
  4. 4
    1. Find the Median (middle value): There are 8 values, so the median is the average of the two middle values (the 4th and 5th). Median = (6 + 8) ÷ 2 = 7.
  5. 5
    1. Find the Mean (sum ÷ count): Sum = 4+5+5+6+8+8+9+10 = 55. Mean = 55 ÷ 8 = 6.875.

Recap

  • The mode is the most common value.
  • The median is the middle value of an ordered dataset.
  • The mean is the sum of values divided by the number of values.
  • The range measures the spread of the data.
  • Always order the data first to find the median and range.
  • A dataset can have more than one mode (bimodal).

Quick check

  1. For the data set {3, 9, 5, 3, 12}, what is the mode?1 mark
  2. For the data set {3, 9, 5, 3, 12}, what is the range?1 mark

3. Organising Data: Frequency Tables

When dealing with large datasets, it's easier to organise the data into a frequency table. This table lists each data value (or group of values) and how many times it occurs (its frequency). For data that is spread out, we use a 'grouped frequency table', where data is sorted into class intervals (e.g., 120-129 cm). From a frequency table, you can find the 'modal class' (the group with the highest frequency) and estimate the mean.

Estimated Mean = Σ(f × x) ÷ Σf

(where x is the midpoint of the class interval and f is the frequency)

Key term

Modal Class: The group or class interval with the highest frequency in a set of grouped data.

Common pitfall

Using the group boundaries (e.g., 60 or 69) instead of the class midpoint (e.g., 64.5) when calculating the estimated mean from a grouped frequency table.

Fun fact

Governments use grouped frequency tables to present census data, for example, showing the population distribution across different age groups.

Worked example 14 marks

The pulse rates of 25 patients were recorded. The results are shown in the grouped frequency table. Find the modal class and estimate the mean pulse rate.

Pulse rate (bpm)Frequency (f)
50-592
60-6911
70-798
80-894
  1. 1
    1. Find the modal class: This is the class with the highest frequency. The highest frequency is 11, which corresponds to the 60-69 bpm group. So, the modal class is 60-69 bpm.
  2. 2
    1. Add columns for midpoint (x) and f × x: The midpoint is the average of the start and end of the interval. For 50-59, midpoint is (50+59)/2 = 54.5.
  3. 3
    Pulse rate (bpm)Frequency (f)Midpoint (x)f × x
    50-59254.5109
    60-691164.5709.5
    70-79874.5596
    80-89484.5338
  4. 4
    1. Find the totals (Σf and Σ(f × x)): Total Frequency (Σf) = 2 + 11 + 8 + 4 = 25. Total of (f × x) column (Σ(f × x)) = 109 + 709.5 + 596 + 338 = 1752.5.
  5. 5
    1. Calculate the estimated mean: Estimated Mean = Σ(f × x) ÷ Σf = 1752.5 ÷ 25 = 70.1 bpm.

Recap

  • Frequency tables show how often each value or group of values occurs.
  • Use grouped frequency tables for continuous or wide-ranging data.
  • The modal class is the group with the highest frequency.
  • To estimate the mean from a grouped table, use the midpoint of each class.
  • The formula for the estimated mean is Σ(f × x) ÷ Σf.

Quick check

  1. In a grouped frequency table, the class '20-29' has a frequency of 15 and the class '30-39' has a frequency of 12. Which is the modal class?1 mark

4. The Probability Scale

Probability measures the chance of an event happening. It is always a value between 0 and 1. An event with a probability of 0 is 'impossible'. An event with a probability of 1 is 'certain'. An event with a probability of 0.5 (or 1/2 or 50%) has an 'even chance' of happening. Probabilities can be written as fractions, decimals, or percentages. The sum of probabilities of all possible outcomes is always 1.

0 ≤ P(event) ≤ 1

P(event will not happen) = 1 – P(event will happen)

Key term

Probability: A measure of the likelihood of an event occurring, expressed as a number between 0 and 1.

Examiner insight

Examiners expect probabilities to be given in their simplest form (e.g., 1/2 instead of 5/10), so always simplify fractions where possible.

Common pitfall

Writing probabilities as values greater than 1 or less than 0. For example, stating a probability is 1.2 or -0.1 is impossible.

Worked example 12 marks

A bag contains 10 balls: 4 are red, 5 are blue, and 1 is green. A ball is picked at random. What is the probability that the ball is NOT blue?

  1. 1
    1. Find the probability of picking a blue ball. There are 5 blue balls out of a total of 10. P(blue) = 5/10 = 1/2.
  2. 2
    1. Use the rule P(not event) = 1 - P(event). The event 'not blue' is the opposite of the event 'blue'.
  3. 3
    1. Calculate the probability: P(not blue) = 1 - P(blue) = 1 - 1/2 = 1/2.
  4. 4
    1. Alternative method: Count the non-blue balls. There are 4 red + 1 green = 5 non-blue balls. P(not blue) = 5/10 = 1/2.

Recap

  • Probability is a number from 0 (impossible) to 1 (certain).
  • Probabilities can be expressed as fractions, decimals, or percentages.
  • An event with a 50% chance is called an 'even chance'.
  • The probability of something not happening is 1 minus the probability that it does happen.

Quick check

  1. The probability of rain tomorrow is 0.3. What is the probability that it will not rain?1 mark

5. Calculating Theoretical Probability

Theoretical probability is what we expect to happen in an ideal situation where all outcomes are equally likely. For example, with a fair six-sided die, each number (1, 2, 3, 4, 5, 6) has an equal chance of being rolled. We calculate it with a simple formula. We also consider 'mutually exclusive' outcomes, which are events that cannot happen at the same time, like rolling a 2 and a 3 in a single throw.

P(event) = (Number of favourable outcomes) ÷ (Total number of possible outcomes)

For mutually exclusive events A and B, P(A or B) = P(A) + P(B)

Key term

Mutually Exclusive: Events that cannot happen at the same time (e.g., a coin toss cannot be both a head and a tail).

Common pitfall

Incorrectly identifying the total number of outcomes, especially in more complex scenarios like rolling two dice.

Worked example 14 marks

A fair spinner has 8 equal sections numbered 1 to 8. What is the probability that the spinner lands on:a) the number 7?b) an even number?c) a prime number?

  1. 1
    1. List all possible outcomes: {1, 2, 3, 4, 5, 6, 7, 8}. Total number of outcomes = 8.
  2. 2
    1. For part (a): There is only one '7'. Number of favourable outcomes = 1. P(7) = 1/8.
  3. 3
    1. For part (b): List the even numbers: {2, 4, 6, 8}. Number of favourable outcomes = 4. P(even) = 4/8 = 1/2.
  4. 4
    1. For part (c): List the prime numbers: {2, 3, 5, 7}. (Remember 1 is not a prime number). Number of favourable outcomes = 4. P(prime) = 4/8 = 1/2.

Recap

  • Theoretical probability assumes all outcomes are equally likely.
  • The formula is P(event) = (favourable outcomes) / (total outcomes).
  • Mutually exclusive events cannot occur simultaneously.
  • Always check the total number of possible outcomes first.
  • Remember to simplify fractions for your final answer.

Quick check

  1. What is the probability of picking a vowel (A, E, I, O, U) at random from the 26 letters of the alphabet?1 mark

6. Experimental Probability & Relative Frequency

Experimental probability, also known as relative frequency, is based on the actual results of an experiment. It's what actually happened, not what was expected to happen. You calculate it by dividing the number of times an event occurred by the total number of trials. The more trials you conduct, the more reliable your experimental probability becomes, and it will generally get closer to the theoretical probability.

Experimental Probability = (Number of times an event occurred) ÷ (Total number of trials)

Key term

Relative Frequency: Another name for experimental probability, calculated from the results of an experiment or observation.

Examiner insight

When asked to compare theoretical and experimental probabilities, students must calculate both values and then make a direct comparison using words like 'higher', 'lower', or 'similar'.

Common pitfall

Confusing the number of successful trials with the total number of trials in the denominator of the calculation.

Fun fact

Weather forecasting uses experimental probability. A '30% chance of rain' means that in the past, under similar atmospheric conditions, it rained 30% of the time.

Worked example 13 marks

A biased die is rolled 200 times. The number '6' appears 50 times.a) What is the experimental probability of rolling a 6?b) If the die is rolled 1000 times, how many 6s would you expect?

  1. 1
    1. For part (a), use the formula for experimental probability. Number of times '6' occurred = 50. Total trials = 200.
  2. 2
    1. P(6) = 50 / 200 = 5/20 = 1/4 (or 0.25).
  3. 3
    1. For part (b), use the experimental probability from part (a) to make a prediction. Expected number = P(event) × Number of new trials.
  4. 4
    1. Expected 6s = (1/4) × 1000 = 250.

Recap

  • Experimental probability is based on results from an experiment.
  • It is also called relative frequency.
  • The formula is (number of successful outcomes) / (total trials).
  • More trials lead to a more reliable estimate of probability.
  • Experimental probability can be used to predict future outcomes.

Quick check

  1. A coin is flipped 50 times and lands on heads 28 times. What is the experimental probability of getting a head?1 mark

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. The following numbers are the ages of 10 people at a bus stop: 23, 19, 45, 23, 67, 18, 34, 45, 23, 50. Find the mean, median, mode, and range of their ages.5 marks
  2. A bag contains 5 red marbles, 3 green marbles, and 2 blue marbles. A marble is picked at random. What is the probability that it is not green?2 marks
  3. A spinner is spun 80 times. The table shows how many times it landed on each colour. What is the experimental probability of landing on Red? Give your answer as a simplified fraction. | Colour | Blue | Red | Yellow | | :--- | :---: | :---: | :---: | | Frequency | 35 | 20 | 25 |2 marks
  4. A school wants to find out if students want a new after-school club. A researcher surveys all the students in the chess club. Explain why this is a biased sampling method.2 marks
  5. A fair 6-sided die is rolled. What is the theoretical probability of rolling a number less than 3?2 marks
  6. The heights of 50 plants were measured. The results are in the table below. Estimate the mean height of the plants. | Height, h (cm) | Frequency | | :--- | :---: | | 0 < h ≤ 10 | 8 | | 10 < h ≤ 20 | 22 | | 20 < h ≤ 30 | 15 | | 30 < h ≤ 40 | 5 |4 marks
  7. The theoretical probability of a biased coin landing on heads is 0.6. How does the experimental probability of getting heads compare to this after 100 flips if it lands on heads 58 times? Is the experimental result higher or lower?3 marks
  8. A two-way table shows the drink choices of 60 people at a cafe. 35 people are adults, and the rest are children. 20 adults chose coffee. 15 children chose juice. In total, 25 people chose juice. Complete a two-way table and find how many children chose coffee.4 marks
  9. The probability of a student passing a test is 4/5. If 150 students take the test, how many would you expect to pass?2 marks
  10. A bag contains only red, blue, and yellow counters. The probability of picking a red counter is 0.2. The probability of picking a blue counter is 0.5. What is the probability of picking a yellow counter?2 marks

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