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An elementary event is an event which contains only a single outcome in the sample space.

For example, if a coin is tossed twice, S = {HH, HT, TH, TT}. Then, {HH}, {HT}, {TH} and {TT} are elementary events.

What is a non-elementary event? Please, give examples.

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An event is a subset of the sample space.

A non-elementary event is an event that is not elementary, i.e. a subset that contains more than (or less than) one element. For example, from the same $S$, you could have $\{HH,TH\}$.

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The events mentioned in your question are elementary because they cannot be written as unions of disjoint non-empty subsets of the sample space.

This in contrast to e.g. $\{HH,TH\}=\{HH\}\cup\{TH\}$ wich is not elementary.

Throwing twice a head is elementary in your example. Throwing a head with the second coin is not.

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