How do you calculate the probability of rolling an even number three times with a six-sided die?

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 formula P(A and B) = $P(A)P(B)$ is crossed out because it is only valid for independent events. The video emphasizes the general multiplication rule, P(A and B) = $P(A)P$(B|A), which works for both independent and dependent events.

Conditions: The events A and B may be dependent.; The general multiplication rule P(A and B) = $P(A)P$(B|A) applies regardless of independence (assuming $P(A) > 0$).

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.

The joint probability P(A and B) is equal to the product of the individual probabilities $P(A)P(B)$ if and only if the events A and B are independent. Independence means that the occurrence of one event does not affect the probability of the other, which is mathematically expressed as P(B|A) = $P(B)$ (assuming $P(A) > 0$).

Conditions: Events A and B are independent.; For the conditional equality P(B|A) = $P(B)$, event A must have positive probability.