each set of probabilities is associated with a rolling cube. What numbers are marked on the faces of each cube?

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

There are six faces on a cube. From the probabilities indicated, 2 sides each have 2, 4 and 6.

To determine the numbers marked on the faces of each cube, we need to consider the probabilities associated with each outcome and the fact that the sum of all probabilities should equal 1.

In this case, we are given three probabilities: P(2) = 1/3, P(4) = 1/3, and P(6) = 1/3.

Let's denote the faces with their respective numbers as a, b, and c. Since the probabilities are equal, we can set up the equation as:

P(a) + P(b) + P(c) = 1

Substituting the given probabilities, we have:

P(a) + P(b) + P(c) = P(2) + P(4) + P(6) = 1/3 + 1/3 + 1/3 = 3/3 = 1

Now, we know that the sum of the probabilities of each face equals 1.

Since each cube has six faces, we can simplify the equation as:

a + b + c = 6

Now we have two equations:

P(a) + P(b) + P(c) = 1
a + b + c = 6

We can solve this system of equations to determine the numbers on each face.

By substituting P(a) with 1/3, P(b) with 1/3, and P(c) with 1/3, we get:

1/3 + 1/3 + 1/3 = 1
a + b + c = 6

Simplifying, we find:

3/3 = 1
a + b + c = 6

Since the sum of each probability is equal to 1 and the sum of each face is equal to 6, we can conclude that each number from 1 to 6 must appear on the faces of the cube. Therefore, the faces of each cube are marked with the numbers 1, 2, 3, 4, 5, and 6.