Rolling two dice – sample space and probability
MathematicsProbabilityAges 14–15
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Probability
The sample space for rolling two dice has 36 equally likely outcomes, arranged in a 6×6 table. Choose an event (sum equals k, sum at most k, both dice equal…), and the favorable cells are highlighted while the theoretical probability P(A) = favorable outcomes / 36 appears instantly. Click to roll N times to compare with the experimental probability; how often each cell occurs is recorded in the table.
Lesson: Random experiments, sample space; probability of an event (rolling-two-dice model)
What it shows
Rolling two fair dice is the classic model of a sample space whose outcomes are ordered pairs. Die 1 and die 2 are distinguishable, so (1; 2) and (2; 1) are different outcomes, and the 36 cells of the table are equally likely. Probabilities are found by counting cells: a sum of 7 fills a whole diagonal of 6 cells, while sums of 2 and 12 have one cell each, so the possible sums are not equally likely. Repeated rolls illustrate the law of large numbers, as the relative frequency settles towards the theoretical probability when N grows.
How to use
Pick an event with the Event buttons and set k, then ask students to predict P(A) before looking at the highlighted cells. Click Roll once a few times, then set Number of rolls N and press Roll N times to compare the Experimental and Theoretical values. Press Reset and repeat with a small and a large N to see how the gap shrinks. Click any cell to check whether that outcome belongs to A.
Parameters you can change
- Event A Sum of pips equals k, Sum of pips at most k, Both dice show the same number, At least one die shows k pips, Product of pips is even
- Value of k 1–12
- Number of rolls N 10–2000
- Roll speed (rolls/frame) 1–10
Questions to explore
- Why are (1; 2) and (2; 1) counted as two different outcomes, and how does this affect P(sum = 3)?
- How many outcomes give an even product, and why is this probability greater than 1/2?
- Which sum is most likely, and how close does the experimental probability get to it after 2000 rolls?
