Two dice sums simulator

What the simulator does

Each roll uses two dice, a white one and a yellow one, and adds them. The sums can be anything from 2 to 12, and the bar chart counts how often each sum comes up. Roll once to see the dice, or roll 10, 100 or 1,000 times at once. Show expected adds a dashed line to each bar for the number you would expect from that many rolls.

The grid lists all 36 outcomes: 6 numbers on the white die times 6 on the yellow die. Tap a bar or a square to light up every outcome with that sum. Find all the ways hides the sums and asks you to tap every pair that makes a target number; when you have them all, the readout gives the probability as a fraction of 36.

Why the bars make a triangle

There is only one way to roll a 2 (1 and 1) but six ways to roll a 7 (1 and 6, 2 and 5, 3 and 4, 4 and 3, 5 and 2, 6 and 1). So 7 has a probability of 6/36, or 1/6, while 2 and 12 have 1/36 each. After a few rolls the chart looks random. After 1,000 it usually looks like a triangle peaking at 7, because the experimental results tend to settle near the theoretical ones. Seeing both the chart and the grid at once connects the experiment to the reason behind it.

Common mix-ups

• "There are 11 sums, so each has a 1 in 11 chance." The sums are not equally likely; the 36 outcomes are.
• "3 and 4 is the same as 4 and 3." With a white and a yellow die you can see they are different outcomes, which is why 7 has six ways, not three.
• "After lots of 7s, a 7 is less likely." Dice have no memory. Every roll is a fresh start.
• "12 is impossible." It is unlikely, 1 in 36, but it happens. Roll 1,000 times and look.

A 10-minute dice lesson

• Ask everyone to pick a prediction for the most common sum. Many will say every sum is equally likely.
• Roll 10 times, then 100. Ask what is happening to the shape.
• Press Find all the ways for a few sums and record the number of ways for each.
• Roll 1,000 times and turn on Show expected.
• Ask: if a game paid out on a sum of 2 or 12, would you play it?

Where it fits by grade

• Grade 7: approximate probability from long-run relative frequency (7.SP.C.6), compare a probability model with observed results (7.SP.C.7), and find probabilities of compound events using organized lists, tables and tree diagrams (7.SP.C.8).
• Grade 6 and below: use the dice as practice with fractions and with reading bar charts.
• England: Key Stage 3 uses tables and grids to list outcomes, generates sample spaces for combined events and calculates theoretical probabilities. Sample spaces continue at GCSE.

Learning more

Fractions and simplifying fractions are skills on Eurekami's skill map, and Daily Practice brings them back when they start to fade. Exam Prep includes GCSE Maths practice papers with probability questions, and the AI tutor asks guiding questions when a problem is tricky.