For two independent events A and B, the multiplication theorem states that:

Concept

The multiplication rule for any two events is P(A ∩ B) = P(A) × P(B | A). It expresses the probability that both events occur by combining a marginal probability with a conditional probability.

When A and B are independent, observing A does not change the probability of B, so P(B | A) = P(B). Therefore the rule reduces to P(A ∩ B) = P(A) × P(B).

Application

Here the events are explicitly independent and “both will occur” means the intersection A ∩ B. Hence their joint probability is the product of their individual probabilities.

Contrast

• For related events, independence cannot be assumed; the conditional term P(B | A) must be retained.

• Adding individual probabilities is associated with union calculations, where overlap must also be accounted for.

• Dividing individual probabilities forms a ratio rather than the joint-event multiplication rule.

• The catch-all statement “None of these” does not apply because the independent-events product statement is present.

Cross-check

Take a fair coin and a fair six-sided die. The events “heads” and “rolling 6” are independent, with probabilities 1/2 and 1/6; the probability of both is 1/12, which equals (1/2) × (1/6).

Result

For independent events A and B, P(A ∩ B) = P(A) × P(B): the probability that both occur equals the product of their individual probabilities.