Two probability problems involving rolling dice

P1. A fair six-sided die is rolled repeatedly until the same number appears on two consecutive rolls. What is the probability that the last number rolled is the same as the first number rolled?

Solution:

\(\dfrac{2}{7}\)

\[ x = \frac{1}{6} + \frac{5}{6} \cdot \frac{1-x}{5}\]

P2. A fair six-sided die is rolled repeatedly until the same number appears on three consecutive rolls. What is the probability that the last number rolled is the same as the first number rolled?

Solution:

\(\dfrac{8}{43}\)

\[ x = \frac{1}{36} + \frac{35}{36} \cdot \frac{1-x}{5}\]