Two things at once
Two dice are rolled together in a board game.
- Die 1 across, die 2 down: 6 × 6 = 36 cells.
- Each cell is one pair, like (3, 4).
- Sum 7 fills 6 cells: (1,6) (2,5) (3,4) (4,3) (5,2) (6,1).
- Chance of sum 7 = 6/36 = 1/6.
Every pair, exactly once
A sum is the two numbers added. Die 1 has 6 faces and die 2 has 6 faces. So two dice make 6 × 6 = 36 pairs. The sums run from 2 to 12, only 11 values. Write die 1 across the top and die 2 down the side. Each cell, or box, holds one pair. (3, 4) means die 1 shows 3 and die 2 shows 4. (3, 4) and (4, 3) are two different pairs.
A die and a coin give 6 × 2 = 12 pairs. One pair is (4, H). H means heads.
The three-results trap
Two coins tossed together give four pairs: HH, HT, TH, TT. "Two heads, one head, or no heads" sounds like three equal results. The three results are not equally likely.
| Coin B: heads | Coin B: tails | |
|---|---|---|
| Coin A: heads | HH | HT |
| Coin A: tails | TH | TT |
For two fair coins, the four pairs are equally likely. "Exactly one head" is HT and TH: two cells of four. So the chance is 2/4 = 1/2. The three-results count gives 1/3, and that is wrong.
"At least one head" means one head or two. It is HH, HT and TH. That is three cells of four. So the chance is 3/4.
Rule: Write the two actions as a table of pairs. Count the pairs an event holds. Divide by all the pairs.
Now use the dice grid. Slide the sum from 2 to 12. The cells that add to it light up. Count them. The chance is that count over 36. You can also tap a cell to pick its sum.
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