Two dice: probability the sum is 9

Two fair dice are rolled simultaneously; find the probability that their sum is 9.

Step-by-step

1. Total outcomes = 6 × 6 = 36
   Two throws, 6 equally likely outcomes each
2. Outcomes summing to 9 = 4
   List the ordered pairs (1,6)(2,5)(3,4)(4,3)(5,2)(6,1)
3. 4/36 ≈ 11.11%
   Probability = favourable outcomes ÷ total outcomes

Why it works

The two dice must be counted as ordered pairs, because rolling a three first and rolling a six first are two distinct equally likely outcomes. There are 36 ordered outcomes in total, and the favorable outcomes must be listed one by one rather than guessed. The numerator of the probability is the number of favorable outcomes, and the denominator is the total number of equally likely outcomes; both must be based on the same sample space.

Common mistakes

• 1) Counting by unordered combinations, the number of outcomes is calculated as 21, making the denominator wrong.
• 2) Treating 11 as the number of possible values for the sum, directly writing it as 4 over 11.
• 3) Omitting symmetric ordered pairs, such as counting only (3,6) but not (6,3).

Related

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