Cambridge Lower Secondary CheckpointStage 8

Statistics and Probability (8Sp)

Mathematics Stage 8 Chapter Notes

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Statistics and Probability - Probability
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In this chapter

  1. 1Organising Data: Tables & Bar Charts
  2. 2Stem-and-Leaf Diagrams and Pie Charts
  3. 3Continuous Data, Time Series & Scatter Graphs
  4. 4Measures of Central Tendency and Spread
  5. 5Comparing Distributions
  6. 6Basic Probability and Complementary Events
  7. 7Experimental Probability (Relative Frequency)
  8. 8Combined Events and Sample Space

1. Organising Data: Tables & Bar Charts

Statistics starts with collecting and organising data. For categorical data (labels, like colours) or discrete data (counted numbers, like shoe size), we often use tables. A tally chart is a simple way to count frequencies. These are then summarised in a frequency table. When you have two categorical variables, a two-way table is excellent for seeing how they relate. To visualise this data, we use bar charts. A dual bar chart places bars side-by-side to compare two datasets, while a compound (or stacked) bar chart stacks bars on top of each other to show how a total is broken down.

Key term

Two-Way Table: A table used to display the frequency distribution of two categorical variables, showing the relationship between them.

Examiner insight

Examiners award marks for correctly labelling axes, using a consistent scale, and clearly distinguishing between bars in dual or compound charts with a key.

Common pitfall

Confusing dual bar charts (which compare two sets of data side-by-side) with compound bar charts (which stack data to show a total).

Worked example 14 marks

80 students were asked if they study French or German. 35 students are boys. 20 of the 48 students who study French are boys.a) Complete the two-way table.b) A student is chosen at random. What is the probability they are a girl who studies German?

  1. 1

    Start by filling in the given numbers: Total students = 80, Total boys = 35, Total French = 48, Boys studying French = 20.

  2. 2

    Calculate Total girls: 80 - 35 = 45.

  3. 3

    Calculate Girls studying French: 48 - 20 = 28.

  4. 4

    Calculate Total German: 80 - 48 = 32.

  5. 5

    Calculate Boys studying German: 35 - 20 = 15.

  6. 6

    Calculate Girls studying German: 45 - 28 = 17. Or 32 - 15 = 17. The table is now complete.

  7. 7

    For part (b), find the number of girls who study German from the table, which is 17.

  8. 8

    The probability is the number of girls studying German divided by the total number of students: P(Girl and German) = 17 / 80.

Worked example 24 marks

The table shows the number of goals scored by two football teams in a season. Draw a suitable chart to compare the data.

Goals0-15 mins16-30 mins31-45 mins46-60 mins61-75 mins76-90 mins
Team A581271015
Team B67991311
  1. 1

    A dual bar chart is suitable for comparing the two teams across the same categories (time intervals).

  2. 2

    Draw and label the axes. The horizontal axis will have the time intervals. The vertical axis will be 'Frequency' or 'Number of Goals' and must have a consistent scale (e.g., 0 to 16 in steps of 2).

  3. 3

    For each time interval, draw two bars side-by-side: one for Team A and one for Team B, up to their respective frequencies.

  4. 4

    For '0-15 mins', draw a bar for Team A up to 5 and a bar for Team B up to 6.

  5. 5

    Repeat this for all six time intervals.

  6. 6

    Include a key to show which colour or shading represents Team A and which represents Team B. Give the chart a title, such as 'Goals Scored by Time Interval for Team A and Team B'.

Recap

  • Tally charts are used for counting raw data.
  • Frequency tables summarise the counts from a tally chart.
  • Two-way tables show the relationship between two categorical variables.
  • Dual bar charts compare two datasets across the same categories.
  • Compound bar charts show how a total is composed of different parts.
  • Always label your charts and axes clearly and include a key where necessary.

Quick check

  1. A two-way table shows that out of 100 people, 60 are adults. 40 people like tea, 25 of whom are adults. How many children do not like tea?2 marks
  2. What type of chart is best for comparing the number of boys and girls who prefer three different sports?1 mark

2. Stem-and-Leaf Diagrams and Pie Charts

Stem-and-leaf diagrams are a concise way to display discrete or continuous data, preserving the original data values. The 'stem' is the leading digit(s) and the 'leaf' is the final digit. They must be ordered and include a key. Pie charts are used to show the proportion of a whole. Each category is represented by a sector of a circle, where the angle of the sector is proportional to the frequency of the category. To calculate the angle, use the formula: Angle = (Frequency / Total Frequency) × 360°.

Pie Chart Angle = (Frequency / Total Frequency) × 360°

Frequency = (Angle / 360°) × Total Frequency

Key term

Stem-and-Leaf Diagram: A diagram that separates each data value into a 'stem' (the first digit or digits) and a 'leaf' (the last digit), organising data while retaining original values.

Examiner insight

Examiners look for a clear, ordered stem-and-leaf diagram. For pie charts, they check that the calculated angles are correct and that the sectors are labelled.

Common pitfall

Forgetting to include a key for a stem-and-leaf diagram, or not ordering the leaves.

Worked example 15 marks

The marks of 15 students in a test were: 27, 34, 45, 29, 38, 45, 27, 21, 30, 38, 49, 41, 38, 23, 33.a) Draw an ordered stem-and-leaf diagram.b) Find the mode and median mark.

  1. 1

    First, draw an unordered diagram to ensure no values are missed. Stems will be 2, 3, 4. Leaves are: 2|7,9,7,1,3; 3|4,8,0,8,8,3; 4|5,5,9,1.

  2. 2

    Now, create an ordered stem-and-leaf diagram by putting the leaves for each stem in numerical order.

  3. 3

    Ordered Diagram: 2 | 1 3 7 7 9; 3 | 0 3 4 8 8 8; 4 | 1 5 5 9.

  4. 4

    Don't forget the key! Key: 2 | 1 means 21.

  5. 5

    b) The mode is the most frequent value. The leaf '8' appears most often in the '3' stem. So, Mode = 38.

  6. 6

    The median is the middle value. There are 15 students, so the middle value is the (15+1)/2 = 8th value. Counting along the leaves, the 8th value is 34. Median = 34.

Worked example 23 marks

A survey of 120 students' favourite subjects showed 45 chose PE, 30 chose Art, 25 chose Maths, and 20 chose Science. Calculate the angle for each sector in a pie chart.

  1. 1

    The total frequency is 120 students.

  2. 2

    Use the formula: Angle = (Frequency / 120) × 360°.

  3. 3

    PE: Angle = (45 / 120) × 360° = 0.375 × 360° = 135°.

  4. 4

    Art: Angle = (30 / 120) × 360° = 0.25 × 360° = 90°.

  5. 5

    Maths: Angle = (25 / 120) × 360° = 75° (approx). Or more accurately (25/120)*360 = (5/24)*360 = 5*15 = 75°.

  6. 6

    Science: Angle = (20 / 120) × 360° = (1/6) × 360° = 60°.

  7. 7

    Check: 135° + 90° + 75° + 60° = 360°. The angles sum to 360°, so the calculations are correct.

Recap

  • A stem-and-leaf diagram must have a key and be ordered.
  • The mode is the most frequent data value; the median is the middle value in an ordered set.
  • Pie charts show proportions of a whole.
  • The total angle in a pie chart is always 360°.
  • To find the angle for a category, divide its frequency by the total frequency and multiply by 360°.

Quick check

  1. In a pie chart, a sector for 'Bus' has an angle of 90°. If 80 people were surveyed, how many travel by bus?2 marks
  2. In the stem-and-leaf diagram with key 5|2 = 52, what values are represented by the row 6 | 0 1 1 8 ?1 mark

3. Continuous Data, Time Series & Scatter Graphs

Continuous data can take any value in a range (e.g., height, time). It is often grouped into classes and displayed in a frequency diagram (or histogram), where the area of the bar represents frequency. Line graphs and time series graphs plot data points over time, connecting them to show trends. A scatter graph is used to investigate the relationship, or correlation, between two numerical variables. By plotting one variable on the x-axis and the other on the y-axis, we can see if there is a pattern. This can be positive correlation (as one increases, the other increases), negative correlation (as one increases, the other decreases), or no correlation.

Key term

Correlation: A measure of the extent to which two variables are related, which can be positive, negative, or non-existent.

Examiner insight

When asked to describe a trend or correlation, examiners expect a clear statement like 'positive correlation' or 'a general upward trend', often with a brief context-based explanation.

Common pitfall

Drawing a line of best fit by just connecting the first and last points, instead of balancing the points on either side of the line.

Worked example 13 marks

The table shows the value of a company's shares at the end of each year.a) Plot a time series graph for this data.b) Describe the trend.

Year201820192020202120222023
Value (£)1.501.701.601.902.102.00
  1. 1

    a) Draw axes. The horizontal axis is 'Year' (2018, 2019, etc.). The vertical axis is 'Value (£)' with a suitable scale (e.g., 1.40 to 2.20).

  2. 2

    Plot each point accurately. For 2018, plot a point at 1.50. For 2019, plot at 1.70, and so on.

  3. 3

    Join the points with straight lines.

  4. 4

    b) To describe the trend, look at the overall direction of the graph. Despite some fluctuations (like the dip in 2020), the overall value of the shares is increasing over the six-year period.

Worked example 23 marks

The scatter graph shows the age and value of ten cars.a) Describe the correlation.b) A car is 7 years old. Draw a line of best fit and use it to estimate its value.

  1. 1

    a) Observe the pattern of the points. As the age of the car (x-axis) increases, its value (y-axis) tends to decrease. This is a negative correlation.

  2. 2

    b) To draw a line of best fit, use a ruler to draw a straight line that passes through the 'middle' of the points, with roughly half the points above the line and half below. The line should follow the direction of the data.

  3. 3

    To estimate the value of a 7-year-old car, find 7 on the x-axis. Move vertically up to your line of best fit. Then, move horizontally to the left to read the corresponding value from the y-axis. The answer will depend on your line, but should be a reasonable value based on the surrounding points.

Recap

  • Continuous data is measured, not counted.
  • Time series graphs show how a variable changes over time.
  • Scatter graphs show the relationship between two numerical variables.
  • Correlation can be positive, negative, or none.
  • A line of best fit can be drawn on a scatter graph to estimate values (interpolation).

Quick check

  1. If ice cream sales increase as temperature increases, what type of correlation is this?1 mark
  2. What type of graph would be best to show the change in your height each year since you were born?1 mark

4. Measures of Central Tendency and Spread

To summarise a dataset, we use measures of 'average' (central tendency) and 'spread' (variation). The three main averages are: the Mean, calculated by adding all values and dividing by how many there are; the Median, the middle value when the data is in order; and the Mode, the most frequent value. The Range is the simplest measure of spread, calculated as the difference between the highest and lowest values (Range = Highest value - Lowest value).

Mean = Sum of all values / Number of values

Range = Highest value - Lowest value

Key term

Central Tendency: A measure that represents the center or typical value of a dataset, such as the mean, median, or mode.

Common pitfall

Forgetting to order the data before finding the median, or calculating the median of an even number of values by just picking one of the two middle numbers.

Fun fact

The mean can be heavily skewed by a single outlier (an unusually high or low value), which is why the median is often a better measure of 'typical' value for things like house prices or salaries.

Worked example 14 marks

Find the mean, median, mode, and range of this dataset: 7, 4, 5, 8, 5, 9, 2.

  1. 1

    First, order the data: 2, 4, 5, 5, 7, 8, 9.

  2. 2

    Mean: Add the values: 2+4+5+5+7+8+9 = 40. There are 7 values. Mean = 40 / 7 ≈ 5.71 (to 2 d.p.).

  3. 3

    Median: The middle value in the ordered list. With 7 values, the middle is the (7+1)/2 = 4th value. The 4th value is 5. Median = 5.

  4. 4

    Mode: The most frequent value. The number 5 appears twice, more than any other. Mode = 5.

  5. 5

    Range: The difference between the highest and lowest value. Range = 9 - 2 = 7.

Worked example 23 marks

The mean height of 5 basketball players is 1.92m. A sixth player with a height of 2.04m joins the team. What is the new mean height of the team?

  1. 1

    First, find the total height of the original 5 players. Total Height = Mean × Number of players = 1.92m × 5 = 9.6m.

  2. 2

    Next, add the new player's height to find the new total height. New Total Height = 9.6m + 2.04m = 11.64m.

  3. 3

    There are now 6 players in the team.

  4. 4

    Calculate the new mean height. New Mean = New Total Height / New number of players = 11.64m / 6 = 1.94m.

Recap

  • The mean is the sum of values divided by the count of values.
  • The median is the middle value of an ordered dataset.
  • The mode is the most frequently occurring value.
  • The range is the difference between the highest and lowest values.
  • Always order the data before finding the median.

Quick check

  1. What is the median of these numbers: 10, 4, 8, 2?2 marks
  2. A dataset has a range of 12 and a maximum value of 20. What is the minimum value?1 mark

5. Comparing Distributions

Comparing two sets of data is a key statistical skill. You can't just list the means and ranges; you must interpret them. To do this, make two separate comparisons: one for the average (using the mean or median) and one for the spread (using the range or interquartile range). A higher mean/median suggests the values in that group are 'on average' higher. A smaller range suggests the values in that group are more consistent, while a larger range suggests they are more spread out or varied.

Key term

Distribution: The way in which data is spread out or clustered across its range of values.

Examiner insight

Top marks are awarded for two distinct comparative statements that are both supported by calculated figures and put into the context of the question.

Common pitfall

Simply stating the calculated values (e.g., 'The mean is 5 and the range is 10') without making a comparative statement using words like 'higher', 'lower', 'more consistent', or 'more spread out'.

Worked example 15 marks

The scores of 10 students in Class A and Class B on a test are shown. Class A: 5, 6, 6, 7, 8, 8, 8, 9, 10, 10. Class B: 2, 4, 6, 7, 8, 8, 9, 9, 10, 10. Calculate the mean and range for each class and compare their performances.

  1. 1

    Calculate for Class A: Mean = (5+6+6+7+8+8+8+9+10+10) / 10 = 77 / 10 = 7.7. Range = 10 - 5 = 5.

  2. 2

    Calculate for Class B: Mean = (2+4+6+7+8+8+9+9+10+10) / 10 = 73 / 10 = 7.3. Range = 10 - 2 = 8.

  3. 3

    Compare the averages: 'On average, Class A scored higher than Class B (mean of 7.7 vs 7.3).'

  4. 4

    Compare the spreads: 'The scores in Class B were more spread out (less consistent) than in Class A, as shown by its larger range (8 vs 5).'

Recap

  • To compare distributions, you must compare both an average and a measure of spread.
  • Use the mean or median to compare the central tendency (the 'average' value).
  • Use the range to compare the spread or consistency of the data.
  • A higher mean means 'higher on average'.
  • A smaller range means 'more consistent'.

Quick check

  1. Team X has a mean goal score of 2.5 and a range of 2. Team Y has a mean of 2.5 and a range of 5. Which team is more consistent?1 mark
  2. If you are comparing two sets of data, what are the two types of measures you must comment on?1 mark

6. Basic Probability and Complementary Events

Probability measures the likelihood of an event happening, on a scale from 0 (impossible) to 1 (certain). It can be written as a fraction, decimal, or percentage. For equally likely outcomes, the theoretical probability of an event A is P(A) = (Number of favourable outcomes) / (Total number of possible outcomes). Complementary events are two outcomes that are the only two possibilities. For example, 'winning' and 'not winning'. The probability of an event not happening, P(not A), is called its complement. The key rule is that the probabilities of an event and its complement always add up to 1.

P(A) = Number of favourable outcomes / Total number of possible outcomes

P(not A) = 1 - P(A)

P(A) + P(not A) = 1

Key term

Complementary Events: Two events that are mutually exclusive and exhaustive, meaning one or the other must occur, so their probabilities sum to 1.

Examiner insight

Examiners expect answers for probability to be given as a fraction in its simplest form, or as a decimal or percentage, unless the question specifies otherwise.

Common pitfall

Assuming that if there are three outcomes (e.g., win, lose, draw), the probability of each must be 1/3. This is only true if they are equally likely.

Worked example 13 marks

A bag contains 5 red, 3 blue and 2 yellow counters. A counter is picked at random. What is the probability that it is:a) blue?b) not blue?

  1. 1

    First, find the total number of counters: 5 + 3 + 2 = 10 counters.

  2. 2

    a) There are 3 blue counters. So, P(blue) = Number of blue counters / Total counters = 3/10.

  3. 3

    b) 'Not blue' means it is red or yellow. There are 5 + 2 = 7 such counters. So, P(not blue) = 7/10.

  4. 4

    Alternatively, using the complement rule: P(not blue) = 1 - P(blue) = 1 - 3/10 = 7/10.

Worked example 21 mark

The probability that a train is late is 0.15. What is the probability that the train is not late?

  1. 1

    The events 'late' and 'not late' are complementary.

  2. 2

    Their probabilities must sum to 1.

  3. 3

    P(not late) = 1 - P(late) = 1 - 0.15 = 0.85.

Recap

  • Probability is measured on a scale from 0 to 1.
  • Theoretical probability is used when all outcomes are equally likely.
  • The sum of probabilities of all possible outcomes is 1.
  • Complementary events are opposites, e.g., 'rain' and 'no rain'.
  • The probability of an event not happening is 1 minus the probability that it does happen.

Quick check

  1. The probability of winning a game is 3/7. What is the probability of not winning?1 mark
  2. A fair six-sided die is rolled. What is the probability of rolling a number less than 3?1 mark

7. Experimental Probability (Relative Frequency)

Sometimes, we can't calculate theoretical probability. How would you know the probability of a drawing pin landing point up? Instead, we perform an experiment. Experimental probability, also known as relative frequency, is an estimate of probability based on the results of an experiment or survey. It is calculated by dividing the number of times an event occurs by the total number of trials. The more trials you conduct, the more reliable the estimate becomes and the closer it tends to get to the true theoretical probability (if one exists).

Experimental Probability = Number of times the event occurred / Total number of trials

Key term

Experimental Probability: An estimate of the likelihood of an event occurring, based on the actual results of an experiment or observation.

Examiner insight

When asked to estimate a number of outcomes, examiners look for the two-step process: first calculating the experimental probability, then multiplying it by the new number of trials.

Common pitfall

Confusing theoretical probability (e.g., P(Heads) = 1/2) with experimental probability (the actual result of flipping a coin, which might not be exactly half).

Worked example 14 marks

A biased spinner is spun 200 times. The results are shown in the table.a) What is the experimental probability of landing on Red?b) If the spinner is spun 500 times, estimate the number of times it will land on Blue.

ColourRedBlueGreen
Frequency845660
  1. 1

    a) The total number of trials is 200. The number of times it landed on Red is 84.

  2. 2

    Experimental P(Red) = Number of Red outcomes / Total trials = 84 / 200.

  3. 3

    Simplify the fraction: 84/200 = 42/100 = 21/50.

  4. 4

    b) First, find the experimental probability of landing on Blue: P(Blue) = 56 / 200 = 7/25.

  5. 5

    To estimate the number of times it will land on Blue in 500 spins, multiply the probability by the number of spins.

  6. 6

    Estimated occurrences = P(Blue) × Number of spins = (7/25) × 500.

  7. 7

    Calculation: 500 / 25 = 20. Then 20 × 7 = 140. We expect it to land on Blue approximately 140 times.

Recap

  • Experimental probability is based on results from trials.
  • It is calculated as (frequency of event) / (total number of trials).
  • It is an estimate, not an exact value.
  • More trials generally lead to a more reliable estimate.
  • To estimate future occurrences, multiply the experimental probability by the new number of trials.

Quick check

  1. A coin is flipped 100 times and lands on heads 47 times. What is the experimental probability of getting tails?1 mark
  2. A dice is rolled 60 times and the number '5' appears 8 times. What is the relative frequency of rolling a '5'?1 mark

8. Combined Events and Sample Space

Many situations involve more than one event, like flipping two coins or rolling two dice. We need systematic ways to list all possible outcomes. A sample space is the set of all possible outcomes. We can represent it using: a simple ordered list; a sample space diagram (a grid); a tree diagram; or a Venn diagram. Once all outcomes are listed, we can calculate probabilities. For independent events (where one event's outcome doesn't affect the other), the probability of both happening is P(A and B) = P(A) × P(B). Tree diagrams are particularly useful for this.

For independent events: P(A and B) = P(A) × P(B)

Key term

Sample Space: The complete set of all possible outcomes of a random experiment.

Examiner insight

For tree diagrams, examiners check that the probabilities on each pair of branches add up to 1 and that the final probabilities are calculated by multiplying along the branches.

Common pitfall

Adding probabilities along the branches of a tree diagram instead of multiplying them.

Worked example 14 marks

Two fair six-sided dice are rolled.a) Show the sample space in a diagram.b) Find the probability that the sum of the scores is 7.c) Find the probability that the score on both dice is the same.

  1. 1

    a) Draw a 6x6 grid. Label the rows 1 to 6 (for Die 1) and the columns 1 to 6 (for Die 2). The cells of the grid represent the outcomes, e.g., (1,1), (1,2), etc. There are 6 × 6 = 36 possible outcomes in total.

  2. 2

    b) To find the sum of 7, look for combinations in the grid: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). There are 6 favourable outcomes.

  3. 3

    P(sum is 7) = Number of ways to get 7 / Total outcomes = 6/36 = 1/6.

  4. 4

    c) The score is the same along the diagonal: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). There are 6 favourable outcomes.

  5. 5

    P(scores are the same) = 6/36 = 1/6.

Worked example 24 marks

The probability of rain on Saturday is 0.8 and on Sunday is 0.3. These events are independent. Draw a tree diagram and find the probability that it rains on exactly one of the days.

  1. 1

    Start the tree diagram with two branches for Saturday: 'Rain' (0.8) and 'No Rain' (1 - 0.8 = 0.2).

  2. 2

    From the end of each Saturday branch, draw two more branches for Sunday: 'Rain' (0.3) and 'No Rain' (1 - 0.3 = 0.7).

  3. 3

    There are four possible outcomes: (Rain, Rain), (Rain, No Rain), (No Rain, Rain), (No Rain, No Rain).

  4. 4

    Calculate the probability of each outcome by multiplying along the branches: P(R, R) = 0.8 × 0.3 = 0.24; P(R, NR) = 0.8 × 0.7 = 0.56; P(NR, R) = 0.2 × 0.3 = 0.06; P(NR, NR) = 0.2 × 0.7 = 0.14.

  5. 5

    'Exactly one day' means (Rain, No Rain) OR (No Rain, Rain).

  6. 6

    Add the probabilities for these two outcomes: P(exactly one day) = 0.56 + 0.06 = 0.62.

Recap

  • A sample space lists all possible outcomes of an experiment.
  • Sample space diagrams (grids) are useful for two events, like rolling two dice.
  • Tree diagrams are useful for sequential events, especially when probabilities are not 50/50.
  • To find the probability of an outcome on a tree diagram, multiply along the branches.
  • If an event can happen in multiple ways, add the probabilities of those separate ways.

Quick check

  1. A coin is flipped and a die is rolled. How many possible outcomes are in the sample space?1 mark
  2. The probability of winning game A is 0.5 and game B is 0.4. What is the probability of winning both?1 mark

End-of-chapter exercise

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

  1. The probability that a biased coin lands on heads is 0.6. The coin is flipped twice. What is the probability that it lands on tails both times?2 marks
  2. The heights (in cm) of 11 plants are: 15, 23, 18, 25, 17, 23, 20, 28, 19, 21, 23. Find the mode, median and mean height of the plants.4 marks
  3. A survey of 200 people's favourite takeaway is shown in a pie chart. The angle for 'Pizza' is 72°. How many people chose Pizza?2 marks
  4. A group of 100 students are asked if they play football or tennis. 60 play football, 35 play tennis, and 15 play both. Represent this information in a Venn diagram and find the number of students who play neither sport.4 marks
  5. The times taken, in minutes, for two groups of runners to complete a 5k race are summarised below. Group A: Median = 28 mins, Range = 12 mins. Group B: Median = 26 mins, Range = 7 mins. Write two sentences to compare the performance of the two groups.2 marks
  6. A bag contains only red, blue and green marbles. The probability of picking a red marble is 1/5. The probability of picking a blue marble is 1/2. If there are 12 green marbles in the bag, how many marbles are there in total?3 marks
  7. A car is tested for its fuel efficiency at different speeds. The results are plotted on a scatter graph. Describe the likely correlation you would expect to see and explain what a line of best fit could be used for.3 marks
  8. An ordered stem-and-leaf diagram shows the ages of people at a party. The row for the 30s is '3 | 1 2 2 5 8'. The median age of all guests is 32. What does this tell you about the distribution of ages at the party?2 marks
  9. A factory produces light bulbs. In a sample of 500 bulbs, 8 were found to be defective. The factory produces 10,000 bulbs a day. Estimate the number of defective bulbs produced in one day.2 marks
  10. Two spinners are spun. Spinner 1 has three equal sections labelled 1, 2, 3. Spinner 2 has four equal sections labelled 1, 2, 3, 4. The scores are added together. By drawing a sample space diagram, find the probability that the total score is an even number.4 marks

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