The Puzzle

You roll three fair six-sided dice in sequence.

Question: What is the probability that the three outcomes come up in strictly increasing order (i.e. first < second < third)?

Total Outcomes

Each die has 6 sides, and they are rolled independently. So the total number of possible outcomes is:

\[ 6 \times 6 \times 6 = 216 \]

Favorable Outcomes

We count the number of strictly increasing sequences. Since all dice must show distinct values in increasing order, we:

  1. Choose 3 distinct numbers from \( {1, 2, 3, 4, 5, 6} \). There are:

\[ \binom{6}{3} = 20 \]

  1. Each such triple has exactly one increasing arrangement (e.g. 2, 4, 6).

So, there are 20 favorable outcomes.

Final Probability

\[ P(\text{strictly increasing}) = \frac{20}{216} = \frac{5}{54} \]

Answer

\[ \boxed{\frac{5}{54}} \]

Reference