Basic Probability: Coins, Dice, Cards

Probability basics

$P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$

Step-by-step: P(sum = 7 with two dice)

Step 1 — Total outcomes: Each die has 6 faces. Total = $6 \times 6 = 36$.

Step 2 — Favorable outcomes (pairs that sum to 7):

$(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$ — that's 6 pairs.

Step 3 — Probability:

$P(\text{sum} = 7) = \frac{6}{36} = \frac{1}{6} \approx 16.67\%$

Why 7 is the most common sum

There are more ways to make 7 than any other sum:

| Sum | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |-----|---|---|---|---|---|---|---|---|----|----|-----| | Ways| 1 | 2 | 3 | 4 | 5 | 6 | 5 | 4 | 3 | 2 | 1 |

The distribution is triangular (symmetric around 7).

Try it in the visualization

The 6×6 grid shows all 36 outcomes. Favorable outcomes for your chosen sum are highlighted. The probability is computed and shown as a fraction and percentage.