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Pairwise Disjoint

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Pairwise disjoint events don’t have any outcomes in common. In probability, the term is often used synonymously with mutually exclusive.

A subtle difference is sometimes defined in set theory. If the intersection of two events is the empty set, then the events are sometimes called pairwise disjoint events. Two events are mutually exclusive if the probability they both happen at the same time (i.e. their union) is zero. By this definition, pairwise disjoint events (which have no outcomes in common) are also mutually exclusive. However, the reverse isn’t always true in set theory, where it is possible to create mutually exclusive events that don’t have the empty set as an intersection.

Some authors don’t differentiate between the two terms at all— not even when it comes to an empty set. For example Olive (2014) gives the following definition (note that Ø is the empty set):

Definition 1.4. If A ∩ B = Ø, then A and B are mutually exclusive or disjoint events. Events A1, A2,…are pairwise disjoint or mutually exclusive
if Ai ∩ Aj = Ø for i ≠ j.

The confusion may stem from the fact that set theory and probability, while very closely connected, are different fields. Each has their own set of naming conventions and definitions. For example, the “universal set (U)” in set theory means the same thing as “sample space (S)” in probability (Olive, 2014). However, they are defined quite differently:

“Definition 2.8 The universal set, at least for a given collection of set theoretic computations, is the
set of all possible objects”~ Ashlock & Lee (2014).

“The set of all the possible outcomes is called the sample space of the experiment and is usually denoted by S”~ Kirk, 2020

Pairwise Disjoint Exhaustive

If the set of all of these non-overlapping events covers the entire sample space then the set of events is pairwise disjoint exhaustive.

As an example (Stewart & Day, 2015), let’s say two parents undergo genetic testing to find out the probability their child will be born with Huntingdon’s disease. The man has the disease and the woman does not. The disease is carried on a particular gene A, so that gives four possibilities for the gene pairs for the parents:


  • Female: aa or Aa
  • Male: AA or Aa

The set of all possible outcomes for this particular event is {child has disease; child does not have disease}. As the events are pairwise disjoint and they cover the entire sample space (i.e. there are no other possible results), they are also exhaustive.


References

Ashlock, D. & Lee, C. (2014). An Introduction to Proofs with Set Theory (Synthesis Lectures on Mathematics and Statistics). Springer.
Kirk, K. Sample Space, Events and Probability. Retrieved October 15, 2020 from: https://faculty.math.illinois.edu/~kkirkpat/SampleSpace.pdf
Olive, D. (2014). Statistical Theory and Inference. Springer.
Stewart, J. & Day, T. (2015). Biocalculus: Calculus, Probability, and Statistics for the Life Sciences. Cengage Learning.

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Stephanie Glen. "Pairwise Disjoint" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/pairwise-disjoint/
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