Question Video: Determining if Two Events are Independent Mathematics

If 𝑃(𝐴) = 0.3 and 𝑃(𝐡) = 0.25 and 𝐴 ∩ 𝐡 = βˆ…, are 𝐴 and 𝐡 independent?

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Video Transcript

If the probability of 𝐴 is 0.3 and the probability of 𝐡 is 0.25 and 𝐴 intersection 𝐡 is equal to the empty set, are 𝐴 and 𝐡 independent?

Let’s begin by considering some of the notation in this question. The intersection notation, written 𝐴 intersect 𝐡, is the set of elements in both set 𝐴 and set 𝐡. 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 𝐡. Recalling our probability formulae, we know that if the probability of 𝐴 intersection 𝐡 is zero, the events are mutually exclusive, whereas when two events are independent, the probability of 𝐴 intersection 𝐡 is equal to the probability of 𝐴 multiplied by the probability of 𝐡.

In this question, since the probability of 𝐴 is equal to 0.3 and the probability of 𝐡 is equal to 0.25, the probability of 𝐴 multiplied by the probability of 𝐡 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 𝐡 is not equal to the probability of 𝐴 intersection 𝐡, the events are not independent. The correct answer is β€œno, they are dependent.”

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

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