Video Transcript
If the probability of 𝐴 is 0.3 and the probability of 𝐵 is 0.25 and 𝐴 intersection 𝐵 is equal to the empty set, are 𝐴 and 𝐵 independent?
Let’s begin by considering some of the notation in this question. The intersection notation, written 𝐴 intersect 𝐵, is the set of elements in both set 𝐴 and set 𝐵. 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 𝐴 and set 𝐵. Recalling our probability formulae, we know that if the probability of 𝐴 intersection 𝐵 is zero, the events are mutually exclusive, whereas when two events are independent, the probability of 𝐴 intersection 𝐵 is equal to the probability of 𝐴 multiplied by the probability of 𝐵.
In this question, since the probability of 𝐴 is equal to 0.3 and the probability of 𝐵 is equal to 0.25, the probability of 𝐴 multiplied by the probability of 𝐵 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 𝐴 multiplied by the probability of 𝐵 is not equal to the probability of 𝐴 intersection 𝐵, the events are not independent. The correct answer is “no, they are dependent.”
If the probability of 𝐴 is 0.3, the probability of 𝐵 is 0.25, and 𝐴 intersection 𝐵 is equal to the empty set, then 𝐴 and 𝐵 are not independent but are dependent.
