Final answer:
The probability that the number 2 will appear on the second roll of the die, given the conditional changes after the first roll, is 1/6.
Step-by-step explanation:
When rolling an unbiased die with the numbers 1, 2, 3, 4, 6, and 8, we need to determine the probability that the number 2 will appear on the second roll after the die changes according to the odd-even rule stated in the problem. The die has three odd (1, 3, 6 replaced by 8) and three even numbers (2, 4, 8). If an odd number is rolled, the odds double, making them 2, 6, and 16 (which we will disregard as they are no longer on the die). If an even number appears on the top face, all the even numbers halve, giving us 1, 2, and 4. The probability that the number 2 will appear on the second roll depends on the result of the first roll.
If the first roll is odd (1, 3, or 8), the die becomes irrelevant for the number 2 in the second roll as the odd numbers double but do not affect the even numbers, so we need not consider those outcomes. However, if we roll an even number (2, 4, 6), the faces of the die become 1 (previously 2), 2 (previously 4), and 4 (previously 8). Now, the number 2 has a 1/3 probability of appearing on the die because there are only three possible outcomes. Therefore, to find the total probability of rolling a 2 on the second roll, we multiply the probability of rolling an even number on the first roll by 1/3. Since there are initially three even numbers out of six sides, the probability of rolling an even number on the first roll is 1/2.
Probability of rolling a 2 on the second roll is: (Probability of rolling an even number on the first roll) x (Probability of 2 on the second roll given the first was even)
= (1/2) x (1/3)
= (1/6).