Question:

Step 1: Identify prime and non-prime faces on a standard die. A standard die has faces numbered from 1 to 6. We classify these numbers into prime and non-prime. The set of all possible outcomes is $\Omega = \{1, 2, 3, 4, 5, 6\}$. Prime numbers on a die are numbers greater than 1 that have no positive divisors other than 1 and themselves. Prime faces: $\{2, 3, 5\}$. There are 3 prime faces. Non-prime faces: $\{1, 4, 6\}$. There are 3 non-prime faces. (Note: 1 is neither prime nor composite). Step 2: Define the probabilities based on the given condition. Let $P(X)$ denote the probability of face $X$ showing up. The problem states that prime number faces are twice as likely to occur as non-prime number faces. Let $P(\text{each non-prime face}) = k$. This means $P(1) = P(4) = P(6) = k$. Then $P(\text{each prime face}) = 2k$. This means $P(2) = P(3) = P(5) = 2k$. Step 3: Use the total probability condition to find the value of $k$. The sum of probabilities of all possible outcomes must be equal to 1. $$P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1$$ Substituting the probabilities in terms of $k$: $$k + 2k + 2k + k + 2k + k = 1$$ Combine like terms: $$(k+k+k) + (2k+2k+2k) = 1$$ $$3k + 6k = 1$$ $$9k = 1$$ Solving for $k$: $$k = \frac{1}{9}$$ Step 4: Determine the individual probabilities for each type of face. Now that we have the value of $k$, we can find the specific probability for each prime and non-prime face. The probability of each non-prime face is $k$: $$P(1) = P(4) = P(6) = \frac{1}{9}$$ The probability of each prime face is $2k$: $$P(2) = P(3) = P(5) = 2 \times \frac{1}{9} = \frac{2}{9}$$ Step 5: Identify the odd number faces and their probabilities. We need to find the probability that an odd number will show up. The odd numbers on a die are $\{1, 3, 5\}$. Let's list the probabilities for these odd faces: $P(1)$: Face 1 is a non-prime face, so $P(1) = \frac{1}{9}$. $P(3)$: Face 3 is a prime face, so $P(3) = \frac{2}{9}$. $P(5)$: Face 5 is a prime face, so $P(5) = \frac{2}{9}$. Step 6: Calculate the total probability of an odd number showing up. The probability of an odd number showing up is the sum of the probabilities of all individual odd outcomes. $$P(\text{odd}) = P(1) + P(3) + P(5)$$ Substitute the individual probabilities calculated in Step 5: $$P(\text{odd}) = \frac{1}{9} + \frac{2}{9} + \frac{2}{9}$$ $$P(\text{odd}) = \frac{1 + 2 + 2}{9}$$ $$P(\text{odd}) = \frac{5}{9}$$ The probability that an odd number will show up is $\frac{5}{9}$. The final answer is $\boxed{\frac{5}{9}}$.