EZ Math

Statistics · Probability

How to Find the Probability of a Sum Less Than 5

Problem

two dice sum less than 5\text{two dice sum less than } 5

Answer

16\frac{1}{6}

Count outcomes, then divide by 36. For “less than,” list every allowed sum before counting pairs.

Step-by-step solution

  1. List the sums under 5.

    2, 3, 42,\ 3,\ 4

    Why: Less than 5 means 2, 3 and 4, not 5 itself.

  2. Count the pairs for each sum.

    1+2+3=6 pairs1 + 2 + 3 = 6\ \text{pairs}

    Why: Sum 2 has one pair, sum 3 has two, and sum 4 has three.

  3. Divide by the 36 total outcomes.

    636=16\frac{6}{36} = \frac{1}{6}

    Why: Two dice have 6 x 6 = 36 equally likely pairs.

Common mistakes

  • Including sum 5 and getting 10/36.
  • Counting only 3 pairs instead of 6.
  • Forgetting to simplify 6/36.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

  1. two dice sum of 3\text{two dice sum of } 3

    Show answer

    118\frac{1}{18}

  2. two dice sum less than 6\text{two dice sum less than } 6

    Show answer

    518\frac{5}{18}

  3. two dice sum greater than 10\text{two dice sum greater than } 10

    Show answer

    112\frac{1}{12}

  4. two dice with an even sum\text{two dice with an even sum}

    Show answer

    12\frac{1}{2}

  5. two dice sum of 4\text{two dice sum of } 4

    Show answer

    112\frac{1}{12}