Event (probability theory)

In probability theory, an event is something that might happen when we do an experiment. For example, when you flip a coin, the experiment has two possible results: heads or tails. An event could be "getting heads" or "getting tails." Events help us talk about what might happen and how likely it is.

An event can be very simple, like getting just one specific result, which is called an elementary event or atomic event. But events can also be more complex, involving many possible results together. These are called compound events. For instance, when rolling a six-sided die, the event "rolling a number greater than 3" includes the results 4, 5, and 6.

We say an event occurs when the result of the experiment matches what the event describes. The chance that an event happens is its probability. Every event has a complementary event, which is simply the event not happening. Together, an event and its complement cover all possible results, making them important for understanding chances in experiments.

A simple example

If we have a deck of 52 playing cards and pick one card, each card is a possible result. An event is any group of these results. For example, an event could be picking a single card, like "The 5 of Hearts," or picking any King, which includes four cards.

Events can also include groups like all Face cards or all cards of a certain suit, such as Spades. When every result is equally likely, the chance of an event happening is found by dividing the number of results in the event by the total number of possible results. This helps us understand probabilities in simple situations.

Events in probability spaces

In probability, an event is a group of possible results from an experiment. When there are only a few possible results, we can look at all the groups. But when there are many possible results, like all numbers on a line, some groups are hard to work with.

To solve this, mathematicians use special collections of groups called σ-algebras. These help us assign probabilities correctly. Only groups inside these collections are called events and have probabilities we can use.

A note on notation

Events in probability are groups of possible results from an experiment. We talk about these events using rules that include random variables. For example, if we have a rule that works with a real-valued random variable, we can describe an event more simply. This makes it easier to work with probabilities and understand chances of different outcomes.