Event (probability theory)

In probability theory, an event is a specific thing that we might observe happening in an experiment. For example, when you flip a coin, the experiment has two possible outcomes: 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 outcome, which is called an elementary event or atomic event. But events can also be more complex, involving many possible outcomes together. These are called compound events. For instance, when rolling a six-sided die, the event "rolling a number greater than 3" includes the outcomes 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 outcomes, making them important for understanding chances in experiments.

A simple example

If we have a deck of 52 playing cards and draw one card, each card is a possible outcome. An event is any group of these outcomes. For example, an event could be drawing a single specific card, like "The 5 of Hearts," or drawing 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 outcome is equally likely, the chance of an event happening is found by dividing the number of outcomes in the event by the total number of possible outcomes. 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 groups. But when there are many possible results, like all numbers on a line, some groups are too strange to handle easily.

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

A note on notation

Events in probability are groups of possible results from an experiment. Even though these events are groups of results, we often write about them using rules that involve random variables. For example, if we have a rule that connects to a real-valued random variable, we can describe an event in a simpler way. This makes working with probabilities easier and helps us understand chances of different outcomes.