Sample spaces for combined experiments
List paired outcomes systematically and count events without overlooking or double-counting results.
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Calculate simple probabilities.
- Read rows and columns of a table.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
How many ordered outcomes are there for two six-sided dice?
What you will learn
- Construct a complete outcome table for two experiments.
- Calculate combined-event probabilities from equally likely pairs.
Make every pair visible
A sample-space table places the outcomes of one experiment in rows and the other in columns. For two fair dice, there are 6 × 6 = 36 equally likely ordered pairs. The pair (2,5) differs from (5,2) because the first and second die are distinguished.
For a sum of seven, the pairs are (1,6), (2,5), (3,4), (4,3), (5,2) and (6,1). Six of the 36 pairs satisfy the event, so the probability is 1/6. Possible sums from two to twelve are not equally likely: some have more supporting pairs.
Only divide by the number of equally likely cells
A coin and a fair three-outcome spinner have six equally likely pairs if the trials are independent and each spinner outcome is equally likely. Listing pairs first prevents assumptions that all event labels have the same chance.
If the component trials are biased or dependent, table cells may not be equally likely. Use their actual probabilities or an appropriate tree. Core combined experiments here use replacement when selecting an item repeatedly from a collection; without-replacement selection is developed as Extended.
Pause and explain
For two fair dice, what is P(sum = 7)?
Worked example
Two fair dice are rolled. Find the probability that their sum is seven.
Show the worked solution
- The sample space contains thirty-six ordered pairs.
- Six pairs have sum seven: (1,6) through (6,1).
- The probability is 6/36 = 1/6.
Answer 1/6.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Two fair dice are rolled. Find P(sum = 2).
Give me a hint
Only one ordered pair gives the smallest sum.
Compare my reasoning
- The favourable pair is (1,1).
- There are thirty-six equally likely pairs.
- Probability is 1/36.
1/36.
Look for these in your work
- Listed the event.
- Used the pair total.
Two fair dice are rolled. Find P(the two scores are equal).
Give me a hint
Count the diagonal cells of the outcome table.
Compare my reasoning
- The equal pairs are (1,1), (2,2), (3,3), (4,4), (5,5), (6,6).
- There are six among thirty-six pairs.
- Probability is 1/6.
1/6.
Look for these in your work
- Counted all equal-score pairs.
- Simplified the probability.
An independent fair coin and spinner labelled 1,2,3 with equal sectors are used. Find P(heads and an odd number).
Give me a hint
List ordered coin–number pairs.
Compare my reasoning
- The six pairs are H1,H2,H3,T1,T2,T3.
- H1 and H3 satisfy the event.
- Probability is 2/6 = 1/3.
1/3.
Look for these in your work
- Built the complete pair list.
- Applied both event conditions.
Common mistakes
- Counting sums as equally likely outcomes.
- Combining reversed ordered pairs into one.
- Ignoring dependence or bias.
What can you explain now?
Are the sums of two fair dice equally likely?
Compare with the explanation
No.
Different sums contain different numbers of equally likely ordered pairs.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
Tomorrow, draw an outcome table and shade the cells satisfying two conditions.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.