Why is $P(A)=m(A)/m(\Omega)$ in geometric probability?

We can multiply the probabilities of individual die rolls because the rolls are assumed to be mutually independent events. For independent events A, B, and C, the probability that all occur is the product of their individual probabilities: $P(A \cap B \cap C) = P(A) \times P(B) \times P(C)$.

Conditions: The events (rolls) must be mutually independent.; Pairwise independence alone is not sufficient for the three-event product rule; mutual independence is required.; The multiplication rule applies to the intersection of independent events.

For a fair six-sided die, the probability of rolling an even number is calculated using the classical probability formula. You count the number of favorable outcomes (even faces) and divide by the total number of possible outcomes (all faces).

Conditions: The die is fair (each face has equal probability).; The sample space is finite and consists of the numbers 1 through 6.; The event of interest is rolling an even number.

To calculate the probability of rolling an even number three times, first find the probability of rolling an even number on a single roll, which is $1/2$ for a fair six-sided die. Then, assuming the three rolls are mutually independent, multiply this single-roll probability by itself three times: $\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$.

Conditions: The die is fair (six-sided, equally likely outcomes).; The three rolls are mutually independent.; The target event is rolling an even number on every one of the three rolls.