Two-way tables (Edexcel GCSE Statistics): Revision Notes

Two-way tables

Two-way tables are an essential tool in probability that help us organise data into rows and columns, making it easier to calculate probabilities and analyse relationships between different categories.

What are two-way tables?

A two-way table displays data by organising it into rows and columns, where each cell shows the frequency (number) of items that belong to both the row category and column category. These tables always include totals for each row and column, plus a grand total representing all the data.

The basic structure includes:

• Rows representing one category (like different types of filling)
• Columns representing another category (like types of bread)
• Row totals showing the sum of each row
• Column totals showing the sum of each column
• Grand total showing the total number of items in the entire dataset

Calculating probabilities from two-way tables

The fundamental probability formula remains the same when working with two-way tables:

Probability = Number of favourable outcomes ÷ Total number of possible outcomes

When using two-way tables, the total number of possible outcomes is usually the grand total (bottom-right cell), whilst the number of favourable outcomes comes from the relevant cell, row total, or column total depending on what you're asked to find.

Step-by-step method for solving two-way table problems

Step 1: Identify what you're looking for

Read the question carefully to understand which probability you need to calculate. Are you looking for:

• A specific combination (one cell)?
• All items in a particular row or column?
• A conditional probability?

Step 2: Find the relevant numbers

• Locate the number of favourable outcomes in the table
• Identify the total number of possible outcomes (usually the grand total)

Step 3: Apply the probability formula

Write the fraction with favourable outcomes on top and total outcomes on the bottom, then simplify if possible.

Step 4: Convert to decimal if required

Divide the numerator by the denominator to get the decimal form.

Worked example: Sandwich shop

Consider a sandwich shop where customers choose different combinations of bread and filling. If we have data showing:

• 10 people chose jam filling out of 32 total customers
• The probability someone chose jam would be: P(jam) = 10/32 = 0.3125

This means there's approximately a 31% chance a randomly selected customer chose jam.

Finding the most likely outcome

When asked to identify the most likely service, product, or outcome, look for the highest total in the relevant category. The category with the largest frequency has the highest probability of being selected.

For example, if comparing different services:

• Service A: 27 occurrences
• Service B: 48 occurrences
• Service B is more likely because it has the higher frequency

Calculating probabilities for specific time periods or categories

Sometimes you'll need to find the probability for a specific month, year group, or category. Use the column or row total that corresponds to your specific category.

For a February probability:

• Count total occurrences in February column
• Divide by grand total
• For example: 30 February occurrences out of 75 total = 30/75 = 2/5

Common exam techniques and tips

Reading the question carefully: Make sure you understand whether you're looking for a single cell, a row total, column total, or combination of cells.

Double-checking totals: Always verify that row totals and column totals are consistent with the grand total - this helps spot errors.

Simplifying fractions: Look for common factors to simplify your probability fractions before converting to decimals.

Conditional probabilities: When the question specifies a particular group (like "a student from Year 10"), use that group's total as your denominator, not the grand total.

Common mistakes to avoid

• Using the wrong total as the denominator
• Misreading which cell contains the favourable outcomes
• Forgetting to simplify fractions
• Mixing up rows and columns
• Not checking that probabilities are between 0 and 1

Remember!

• Two-way tables organise data into rows and columns with totals for easy probability calculations • Always use the formula: Probability = Favourable outcomes ÷ Total possible outcomes
• The grand total (bottom-right corner) is usually your total number of possible outcomes • The most likely outcome is the one with the highest frequency • Double-check your work by ensuring all totals add up correctly