Measurement, Space & Statistics
Bivariate Data and Conditional Probability
Year 10
- ✓By the end of this lesson students will be able to identify and describe bivariate data.
- ✓By the end of this lesson students will be able to construct and interpret scatter plots to identify associations between two numerical variables.
- ✓By the end of this lesson students will be able to understand and apply the concept of conditional probability.
- ✓By the end of this lesson students will be able to calculate conditional probabilities using two-way tables and the conditional probability formula.
Key concepts
Bivariate data involves two different variables that are measured for each individual or item. The purpose of collecting bivariate data is often to investigate if there is a relationship or association between the two variables. For example, measuring a student's height and their maths test score to see if there's a relationship.
A scatter plot is a graph used to display bivariate numerical data. Each point on the scatter plot represents a pair of data values for a single individual. By examining the pattern of points, we can describe the association (or relationship) between the two variables. We look for:\n- **Direction:** Positive (as one variable increases, the other tends to increase), Negative (as one variable increases, the other tends to decrease), or No association.\n- **Form:** Linear (points tend to follow a straight line) or Non-linear.\n- **Strength:** Strong (points are tightly clustered around a line/curve), Moderate, or Weak (points are widely scattered).
Conditional probability is the probability of an event occurring given that another event has already occurred. It's about how the probability of an event changes when we have extra information. We use the notation P(A|B), which is read as 'the probability of event A occurring given that event B has occurred'. The 'given that' part restricts our sample space to only those outcomes where event B has occurred.
Two-way tables (also known as contingency tables) are extremely useful for organising categorical bivariate data and for calculating probabilities, especially conditional probabilities. The rows and columns represent the categories of two different variables, and the cells contain the frequencies (counts) of observations that fall into both categories. Total rows and columns provide marginal frequencies, and the grand total is the total number of observations.
Key facts to remember
- 1Bivariate data involves two variables, often to explore relationships.
- 2Scatter plots display bivariate numerical data, showing direction, form, and strength of association.
- 3A positive association means as one variable increases, the other tends to increase.
- 4A negative association means as one variable increases, the other tends to decrease.
- 5Conditional probability, P(A|B), is the probability of A occurring given that B has already occurred.
- 6The formula for conditional probability is P(A|B) = P(A ∩ B) / P(B), where P(B) ≠ 0.
- 7Two-way tables are excellent tools for visualising and calculating probabilities, especially conditional probabilities, from categorical data.
Worked examples
Example 1
A study investigated the relationship between the number of hours a student spent studying for a maths exam and their score on the exam. The data is plotted on a scatter plot. Describe the association between study hours and exam score.
Answer
The scatter plot shows a **positive, linear, and strong association** between the number of hours studied and the exam score. As the number of hours studied increases, the exam score generally increases, and the points are quite close to forming a straight line.
Always describe direction, form, and strength when interpreting associations from scatter plots.
Example 2
A survey of 200 students asked if they preferred maths or English. The results are shown in the two-way table below:\n\n| | Prefers Maths | Prefers English | Total |\n|-----------|---------------|-----------------|-------|\n| **Year 10** | 50 | 30 | 80 |\n| **Year 11** | 60 | 60 | 120 |\n| **Total** | 110 | 90 | 200 |\n\na) What is the probability that a randomly selected student prefers English?\nb) What is the probability that a randomly selected student is in Year 10 and prefers Maths?\nc) Given that a student prefers English, what is the probability that they are in Year 11?
Answer
a) P(Prefers English) = 0.45\nb) P(Year 10 ∩ Prefers Maths) = 0.25\nc) P(Year 11 | Prefers English) = 2/3 or approximately 0.667
For conditional probability from a two-way table, the denominator is the total of the 'given' event, not the grand total.
Example 3
In a particular school, 60% of students play a sport (S) and 30% play a musical instrument (M). 20% of students play both a sport and a musical instrument. What is the probability that a student plays a sport given that they play a musical instrument?
Answer
The probability that a student plays a sport given that they play a musical instrument is 2/3.
Ensure you correctly identify P(A ∩ B) and P(B) for the formula P(A|B).
Common mistakes
- ✗Confusing P(A|B) with P(B|A). The order matters significantly.
- ✗Using the grand total as the denominator for conditional probability calculations when working with two-way tables. The denominator should be the total of the 'given' event.
- ✗Misinterpreting the direction of association on a scatter plot (e.g., calling a negative association positive).
- ✗Not clearly identifying the 'intersection' (A ∩ B) when applying the conditional probability formula.
- ✗Assuming causation from correlation. A strong association does not necessarily mean one variable causes the other.
Exam tips
- ★Always read 'given that' questions carefully to correctly identify the condition (event B) and the event whose probability is being sought (event A).
- ★When working with two-way tables, highlight or circle the row/column corresponding to the 'given' event to clearly define your reduced sample space.
- ★For scatter plots, remember to describe the association in terms of direction (positive/negative/none), form (linear/non-linear), and strength (strong/moderate/weak).
- ★Practise using both the formula and two-way tables for conditional probability problems, as different problems might be better suited to one method over the other.
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