Question Video: Finding the Probability of Union of Complement of Two Mutually Exclusive Events Mathematics

If 𝐴 and 𝐡 are two mutually exclusive events from a sample space of a random experiment, find 𝑃(𝐴 complement ⋃ 𝐡 complement).

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Video Transcript

If 𝐴 and 𝐡 are two mutually exclusive events from a sample space of a random experiment, find the probability of 𝐴 complement union 𝐡 complement.

To answer this question, we need to recall one of De Morgan’s laws: 𝐴 complement union 𝐡 complement is equivalent to the complement of 𝐴 intersect 𝐡. And so, it follows that the probability of 𝐴 complement union 𝐡 complement will be the probability of the complement of 𝐴 intersect 𝐡. Now, this law holds regardless. But these two events in this question are mutually exclusive. And we know that for mutually exclusive events, their intersection is the empty set. This means that the complement of their intersection is the entire sample space, which has a probability of one.

Or more formally, we can say that the probability of 𝐴 intersect 𝐡 complement is one minus the probability of 𝐴 intersect 𝐡. That’s one minus zero, which is one. So, by recalling De Morgan’s law and that for mutually exclusive events the probability of their intersection of zero, we’ve found that for the two mutually exclusive events 𝐴 and 𝐡, the probability of 𝐴 complement union 𝐡 complement is one.

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