 # Event Space (Probability): Simple Definition & Examples

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An event space contains all possible events for a given experiment or happening. An event is just a set of outcomes of an experiment, combined with their probability. And an “experiment” has a meaning a little different to the one we use in ordinary speech: it’s anything, with a well-defined set of possible outcomes, that can be repeated infinitely.

## Examples of Event Space

An event space is often denoted by the Greek letter sigma (Σ). It’s sometimes confused with the sample space of an experiment, referred to usually by omega(Ω), but is different: while the sample space of an experiment contains all possible outcomes, the event space contains all sets of outcomes; all subsets of the sample space.

## Example 1: Coin Flip

For a simple coin flip, the two possible outcomes are either heads or tails, so the sample space is given by

Ω = {H, T}

The event space is a little different. The possible events are:

• {H}—rolling the die and getting heads,
• {T}—rolling the die and getting tails,
• {H,T}—rolling the die and getting either heads or tails.

Because each of these are different subsets of the sample space, they count as different events, even though {H} (heads) would imply {H, T} (either H or T). The event space contains all three of those events:

Σ= {(H), (T), (H,T)}

## Example 2: Coin Flip

For the roll of a die, the sample space is {1, 2, 3, 4, 5, 6}.

One event, if you roll the die three times, is {2,4,6}

(In English, we’d call this ‘the roll of the dice is even’)

Another event could be {1,2}

(We could call this ‘getting less than 3’)

The collection of every possible set (there are an infinite amount, depending on how many times you choose to roll that die) of those six numbers form the event space Σ.

## Sources

Kirkpatrick, K. Sample Space, Events, and Probability