Mathematics 7–10 · Year 9
Sum of two dice: 36 equally likely pairs, 11 unequal totals
Statistics and probability
This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.
The idea
A two-stage experiment has a sample space of ordered pairs; counting the pairs that give each total explains why 7 is the most frequent sum and 2 and 12 the rarest.
What you need
- 2 dice of different colours per pair (so (2, 5) and (5, 2) are seen as different outcomes)
- 1 shaker cup and tray per pair
- Tally sheet with rows for totals 2 to 12
- Grid paper for the 6 by 6 array of outcomes
How to do it
- Predict which total will come up most often in 36 rolls and how often.
- Draw the 6 by 6 array of ordered pairs and write the total in each cell; count how many cells give each total (1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1).
- Roll both dice 36 times and tally the totals. Compare the tally with the counts in the array.
- Pool the class (10 pairs gives 360 rolls) and draw a bar chart of the pooled frequencies beside the theoretical bar chart of k/36.
- Use the array to find the probabilities of events such as a total of 8 or more (15 cells out of 36) and a double (6 cells out of 36).
- Test the claim that odd and even totals are equally likely by counting cells (18 each).
What you should see
The array gives P(7) = 6/36 = 0.1667 and P(2) = P(12) = 1/36 = 0.0278. In 360 pooled rolls the expected number of sevens is 60 with a standard deviation of 7.07, so 46 to 74 sevens is ordinary; the expected number of twos is 10. The pooled bar chart takes the triangular shape of the theoretical chart while a single pair's 36 rolls looks ragged. Odd totals occur 18 times out of 36 pairs, so P(odd) = 0.5 even though only five of the eleven totals are odd. The learner knows it worked when the pooled sevens fall inside 46 to 74 and the triangular shape is visible.
What changes
- What you change
- number of rolls
- What you measure
- frequency of each total
- What you keep the same
- same two dice
- same rolling method
- both dice read every time
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- Every total from 2 to 12 is equally likely.
- A total of 3 has one way (1 and 2) rather than two ordered ways.
- Even totals are more likely because there are six of them and five odd ones.
Safety card
Hazards
- dice dropped on the floor
Controls
- roll into a tray
Note
No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics K–10 Syllabus (2022), Stage 5 Probability A; page read 2026-09-22MA5-PRO-C-01
- Australian Curriculum v9AC9M9P01
Sources
The pages the author read to write this activity.