A single dle is rolled twice. Find the probability of roiling a 6 the first time and a 6 the second time, Find the probabilly of roilling a 6 the first time and a 6 the second time. \( \square \) (Type an integer or a simplified fraction.)

When you roll a die, the probability of rolling a 6 is \( \frac{1}{6} \). Since the rolls are independent, the probability of rolling a 6 on the first roll and a 6 on the second roll is given by multiplying the probabilities of each event. Therefore, the calculation is: \[ P(\text{6 on first roll}) \times P(\text{6 on second roll}) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \] So, the probability of rolling a 6 the first time and a 6 the second time is \( \frac{1}{36} \). Rolling dice has a rich history dating back to ancient civilizations. The oldest known dice date from around 3000 BC in the Indus Valley and are made from materials like bones and ivory. Fast forward to modern times, and we see dice being used not just in games but also in statistical experiments and probability theory, showcasing their significance in both fun and academic realms! In practice, understanding these probabilities can help in various situations, from playing board games to wagering on games of chance. A common mistake is assuming that past rolls influence future outcomes, known as the gambler's fallacy. Each roll is independent, so the chances stay the same regardless of previous results. Embracing this concept can improve your game strategies significantly!