Two dice, 36 pairs
You throw two dice. Each die has faces 1 to 6, all with the same chance. You add the two numbers to get the sum. Is a sum of 2 as likely as a sum of 7?
Step 1 of 4
- Die 1 across, die 2 down. 6 × 6 = 36 cells.
- Sum 7 is in 6 cells: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1).
- Sum 2 is in just 1 cell: (1, 1).
- P(sum 7) = 6/36 = 1/6. P(sum 2) = 1/36. Not equal!
No. A sum of 7 is more likely than a sum of 2. It fits 6 of the 36 pairs. A sum of 2 fits 1.
Count pairs, not sums
The outcomes are the 36 pairs. Each pair is equally likely, because every face has the same chance. The sums are 2 to 12, which is 11 sums. The 11 sums are not equally likely. Some sums fill more cells.
| Sum | Cells | Chance |
|---|---|---|
| 2 | 1 | 1/36 |
| 7 | 6 | 6/36 = 1/6 |
| 8 | 5 | 5/36 |
| 11 | 2 | 2/36 = 1/18 |
| 12 | 1 | 1/36 |
The same grid answers other questions too.
- Same number on both dice (a doublet): 6 cells. P = 6/36 = 1/6.
- Die 1 bigger than die 2: 15 cells. P = 15/36 = 5/12.
Now try it. Slide the sum. Count the lit cells.
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