What would you like to understand?

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.

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.

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.

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.

The assumption used is that the rolls are mutually independent. This means that the outcome of one roll does not affect the probability of the outcome of any other roll.

Conditions: The model assumes mutual independence of the three rolls.; Fairness of the die alone does not establish independence.; The probability of an event on one roll is unaffected by events on other rolls.

The assumption used is that the rolls are mutually independent. This means that the outcome of one roll does not affect the probability of the outcome of any other roll.

Conditions: The model assumes mutual independence of the three rolls.; Fairness of the die alone does not establish independence.; The probability of an event on one roll is unaffected by events on other rolls.