Rolling Sixes in a Row

Solution & Explanation

To solve this problem, we need to find the expected number of rolls required to achieve six consecutive sixes for the first time when rolling a standard six-sided die. This involves calculating the expected value of a random variable that counts the number of rolls needed to achieve a specific sequence.

Conceptual Approach

1. Define the Event:

   • Let $E$ be the expected number of rolls needed to get six consecutive sixes.
   • The probability of rolling a six on a single roll is $\frac{1}{6}$.

2. Break Down the Problem:

   • Case 1: The first roll is not a six.

     • Probability: $\frac{5}{6}$
     • Expected rolls: $E + 1$

   • Case 2: The first roll is a six, but the second roll is not a six.

     • Probability: $\frac{1}{6} \times \frac{5}{6}$
     • Expected rolls: $E + 2$

   • Continue this pattern until:

   • Case 6: Six consecutive rolls are all sixes.

     • Probability: $\left(\frac{1}{6}\right)^6$
     • Expected rolls: 6 (since we succeed here)

3. Formulate the Equation:

   The expected number of rolls $E$ can be expressed as:

   $E = \frac{5}{6}(E + 1) + \frac{1}{6} \times \frac{5}{6}(E + 2) + \ldots + \left(\frac{1}{6}\right)^6 \times 6$

4. Solve the Equation:

   • Simplifying the equation involves recognizing the pattern and solving for $E$.

   • A closed-form solution can be derived using the formula for the expected number of trials until $n$ consecutive successes:

     $E = \frac{1 - p}{p^n - 1}$

     where $p = \frac{1}{6}$ and $n = 6$.

   • Plugging in the values:

     $E = \frac{1 - \frac{1}{6}}{\left(\frac{1}{6}\right)^6 - 1}$

   • Calculating:

     $E \approx 46656$

Conclusion

The expected number of rolls needed to achieve six consecutive sixes is approximately 46656 rolls. This result illustrates the rarity of achieving such a sequence due to the low probability of rolling a six consecutively six times.