Two dice: probability the sum is 4

Two fair dice are rolled. What is the probability that the sum of the two numbers is 4?

Step-by-step

1. Total outcomes = 6 × 6 = 36
   Two throws, 6 equally likely outcomes each
2. Outcomes summing to 4 = 3
   List the ordered pairs (1,6)(2,5)(3,4)(4,3)(5,2)(6,1)
3. 3/36 ≈ 8.33%
   Probability = favourable outcomes ÷ total outcomes

Why it works

Each ordered pair of results is equally likely, giving 6 × 6 = 36 outcomes in total. Counting the pairs whose sum is 4 gives the favourable outcomes, and the probability is that count divided by 36.

Common mistakes

• Counting only 1+3 and 3+1 style pairs as one outcome instead of two.
• Dividing by 12 (the number of dice faces) instead of by 36.

Related

dice-probability dice-sum-five-probability n-math-probability-two-dice-sum