Final Answer:
a. For the probability of rolling a 6 on the 3rd roll: 5/216
b. For the probability of not rolling a 6 until after the 5th roll: 5/6
Step-by-step explanation:
a. To calculate the probability of rolling a 6 on the 3rd roll of a fair six-sided die, we recognize that there are five non-6 outcomes in the first two rolls (5/6 probability for each roll), followed by a 1/6 probability of rolling a 6 on the third roll. Therefore, the probability is (5/6) * (5/6) * (1/6) = 5/216.
b. To find the probability of not rolling a 6 until after the 5th roll, we consider the complementary probability of rolling a 6 in the first five rolls. The probability of not rolling a 6 in a single roll is 5/6, and for five consecutive rolls, it is (5/6)^(5). Subtracting this probability from 1 gives the probability of rolling a 6 after the 5th roll, which is 1 - (5/6)^(5) = 5/6.
Understanding probabilities in dice rolls is fundamental in probability theory and has applications in various fields, including statistics and game theory.
Exploring probability theory provides insights into understanding and predicting outcomes in random events, contributing to informed decision-making in diverse fields.