Final answer:
The probability of rolling a 1 or a 6 on the loaded die is 50%. This is calculated by assigning the probability of rolling a 1 as 1/8, and since 6 appears three times as often, its probability is 3/8. Adding both probabilities gives us the combined probability of rolling a 1 or 6 (0.5).
Step-by-step explanation:
The student is asking about the probability of rolling a 1 or a 6 on a loaded die where the number 6 appears three times more often than any other number. To calculate this probability, we must consider the die's altered probabilities due to it being loaded. Normally, each number on a fair six-sided die has an equal probability of ⅖ or approximately 0.1667. However, since the die is loaded to favor the number 6, we need to adjust the probabilities.
Let's define the probability of rolling a 1 as P(1), which would remain at ⅖ under normal circumstances. Since the number 6 comes up three times as often as the other numbers, we can define the probability of rolling a 6, as P(6) = 3 × P(1). The total probability from all six sides of the die must add up to 1 (the certainty of rolling some number). So, we have P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1, with each probability equal to P(1) except P(6), which is 3 × P(1). We, therefore, have 5 × P(1) + 3 × P(1) = 1, leading to P(1) being ⅛.
To find the probability of rolling a 1 or a 6, we add the individual probabilities of each event: P(1) + P(6). Using the value of P(1) we found, we get P(1) = ⅛, and P(6) = 3 × ⅛ = ¾. Thus, P(1 or 6) = ⅛ + ¾ = ½ or 0.5, which means there is a 50% chance of rolling either a 1 or a 6 on this loaded die.