### Video Transcript

If the probability of π΄ is 0.3 and the probability of π΅ is 0.25 and π΄ intersection π΅ is equal to the empty set, are π΄ and π΅ independent?

Letβs begin by considering some of the notation in this question. The intersection notation, written π΄ intersect π΅, is the set of elements in both set π΄ and set π΅. We are told in this question that this is equal to the empty set, and this means there are no elements that are in both set π΄ and set π΅. Recalling our probability formulae, we know that if the probability of π΄ intersection π΅ is zero, the events are mutually exclusive, whereas when two events are independent, the probability of π΄ intersection π΅ is equal to the probability of π΄ multiplied by the probability of π΅.

In this question, since the probability of π΄ is equal to 0.3 and the probability of π΅ is equal to 0.25, the probability of π΄ multiplied by the probability of π΅ is equal to 0.3 multiplied by 0.25, and this is equal to 0.075. We can therefore conclude that since the probability of π΄ multiplied by the probability of π΅ is not equal to the probability of π΄ intersection π΅, the events are not independent. The correct answer is βno, they are dependent.β

If the probability of π΄ is 0.3, the probability of π΅ is 0.25, and π΄ intersection π΅ is equal to the empty set, then π΄ and π΅ are not independent but are dependent.