Problem 6 · Count arrangements and ties · 4 points
Made by AI and not reviewed by a real person.
Three different-colored fair six-sided dice are rolled. What is the probability that at least one die shows the average of the other two?
- 1/4
- 7/36
- 1/36
- 1/12
- 1/6
Show a hint
- Try sorting the three rolls. What successful patterns are possible, including when numbers match?
- For three distinct rolls, look for an equally spaced triple. Then count its arrangements on the colored dice separately from the all-equal rolls.
Show the answer and solution
Answer: B — 7/36
The distinct triples that work are (1,2,3), (2,3,4), (3,4,5), (4,5,6), (1,3,5), and (2,4,6). Each has 6 arrangements on the colored dice, giving 36 outcomes. The 6 all-equal outcomes also work, for a total of 42 out of 216 equally likely outcomes. Thus the probability is 42/216 = 7/36. The tempting 1/6 counts the distinct triples but misses the all-equal rolls.