Statistics How To

Intersection of Events: Definition & Examples

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An Intersection is a collection of elements (like people, places, or things) that belong to two or more sets.

a union b

A Venn diagram showing household objects that could be classified as either electronic or non-electronic. The intersection contains toothbrushes and lawnmowers, which belong to both sets.

The intersection is like a container with the elements as the contents. The container itself says nothing about the probability that an intersection of two (or more) events will happen. For that, you’ll need to calculate the joint probability.

If the events don’t have any outcomes in common, they are called pairwise disjoint or mutually exclusive events. The two terms are sometimes used to mean the same thing, but there is an important difference in set theory:

  • Pairwise disjoint events have an intersection that is the empty set.
  • Mutually exclusive events have a zero probability of happening at the same time.

How to Draw an Intersection (A ∩ B)

Visually, items in an intersection appear in both categories at the same time.

Example question: Draw the intersection of the following black and white objects:

  • Cats,
  • Coal,
  • Hair,
  • Milk,
  • Onyx,
  • Shaving Cream,
  • Snow,
  • Tar.

Step 1: Place the items into categories (in this case, “black things” and “white” things):

  • Black things: Cats, Coal, Hair, Tar, Onyx.
  • White things: Cats, Hair, Snow, Milk, Shaving Cream.

Step 2: Draw two or three circles (i.e. a Venn diagram) according to how many categories you have. Make sure the circles overlap, because that’s where the intersection is.

Step 3: Write the items that the lists have in common (i.e. those that are in the intersection) in the space where the circles overlap.
For this example, that’s {cats, hair}.

Step 4: Fill in all of the other items to complete the Venn diagram.

That’s it!
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Agresti A. (1990) Categorical Data Analysis. John Wiley and Sons, New York.
Gonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial.
Vogt, W.P. (2005). Dictionary of Statistics & Methodology: A Nontechnical Guide for the Social Sciences. SAGE.

Stephanie Glen. "Intersection of Events: Definition & Examples" From Elementary Statistics for the rest of us!

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