Expert: Socrates - 1/24/2004

Question
If three fair standard dice are tossed, what is the probability that the results will be three consecutive integers?

Answer
There are 6x6x6 = 216 possible outcomes when the three dice are tossed. The three consecutive integers can be
1,2,3
2,3,4
3,4,5
4,5,6

Each of these sequences can come up in 6 ways. Consider the  sequence 4,5,6 . To get this sequence, there are 3 possible numbers for the first die , two for the second, and one for the third. This means there are 6 ways to get 4,5,6. The same is true for the other sequences of consecutive integers.
We now have 4 possible sequences, and each can occur in 6 ways, giving a total of 6x4 = 24 possible outcomes of three consecutive integers.
The probability is then 24/216 = 12/108 = 6/54 = 1/9.
The answer is 1/9.