07 / 10 · Probability
Two dice
Why 7 is the most common sum
Roll two dice and add them. The sum can be anything from 2 to 12, but those eleven sums are not equally likely.
The trick is to count pairs, not sums. The first die has 6 faces and so does the second, so there are 6 × 6 = 36 equally likely pairs. A sum of 2 needs both dice to show 1: only one pair. A sum of 7 can be 1+6, 2+5, 3+4, 4+3, 5+2 or 6+1: six pairs.
36 pairs, all equally likely P(sum = 7) = 6 ÷ 36 ≈ 0.167 P(sum = 2) = 1 ÷ 36 ≈ 0.028
Try it · Every cube is one pair of dice, stacked above its sum. Pick a sum to highlight its pairs, or roll the dice and see which column you land in.
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So P(sum = 7) = 6/36 = 1/6, while P(sum = 2) = 1/36. Seven is six times more likely than two.
Stack the pairs by their sum and you get a triangle. Add more dice and the triangle rounds off into a bell shape, which is where this course is heading.
Key ideas
Words worth remembering
- Independent events
- Events where one outcome does not affect the other, like two separate dice.
- Multiplication rule
- The number of combined outcomes is the product: 6 × 6 = 36.
- Distribution
- How probability is spread across all possible results.
Think about it
Which is more likely with two dice: a sum of 6 or a sum of 8?