a die is rolled in such a way that even faces are twice as likely to occur as the odd faces. what is the probability that a prime number shows up when rolled?

since odds come up 1/3 of the time, I guess it depends on how many of the odd faces are primes. You haven't said how many faces are on the die.

anyway, since 2/3 of the odd numbers on a regular die are primes, that means that a prime will come up 2/9 of the time.

well they have to add up to one and there are 6 sides

p(1) = n prime
p(2) = 2n prime
p(3) = n prime
p(4) = 2n
p(5) = n prime
p(6) = 2n

sum of p s is one
9 n = 1
n = 1/9
primes are 5n or 5/9

To find the probability that a prime number shows up when rolling the die, we first need to determine the probabilities of each outcome.

Let's denote the probability of an even face as P(E) and the probability of an odd face as P(O). We know that even faces are twice as likely to occur as odd faces. Mathematically, we can represent this as:

P(E) = 2 * P(O)

Since a standard die has 6 faces, we can divide the probabilities equally among them. Let's denote the probability of rolling each face as P(1), P(2), P(3), P(4), P(5), and P(6).

Since there are three even faces (2, 4, and 6), and they are twice as likely to occur as the odd faces, we can calculate the probability of an even face as:

P(E) = P(2) + P(4) + P(6) = (1/6) + (1/6) + (1/6) = 3/6 = 1/2

Using the relationship we established earlier, we can calculate the probability of an odd face:

P(O) = P(E) / 2 = (1/2) / 2 = 1/4

Now, let's calculate the probability of rolling a prime number (2, 3, or 5). Since each face has an equal probability, we can sum the probabilities of rolling each prime number:

P(prime) = P(2) + P(3) + P(5)

P(2) = P(E) / 3 = (1/2) / 3 = 1/6

P(3) = P(O) / 2 = (1/4) / 2 = 1/8

P(5) = P(O) / 2 = (1/4) / 2 = 1/8

Therefore,

P(prime) = P(2) + P(3) + P(5) = 1/6 + 1/8 + 1/8 = 4/24 + 3/24 + 3/24 = 10/24

The probability of rolling a prime number when using a die rolled in such a way that even faces are twice as likely to occur as the odd faces is 10/24, which simplifies to 5/12.