Theoretical and experimental probability, complementary events, expected number of outcomes, sample space
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In probability, a trial is any procedure with an uncertain result, such as flipping a coin, rolling a die, or drawing a card. Each possible result of a trial is called an outcome.
An event is a collection of one or more outcomes, representing scenarios we're interested in, such as rolling an even number or drawing a red card. Events are the probabilities we calculate, and are typically denoted with letters such as A, so that the "probability of an event A" is given by P(A).
All possible outcomes from a single trial form the sample space, denoted U.
The overall probability of the sample space, denoted P(U), is 1. This expresses the idea that if you perform a trial, something must happen.
Theoretical probability is calculated based on reasoning or mathematical principles—it's what we expect to happen. When outcomes are equally likely, the probability of an event is given by
where n(A) is the number of outcomes in event A, and n(U) is the total number of outcomes in the sample space.
A sample space diagram is a table whose cells consist of all possible outcomes in a given sample space. The probability of an event A in the diagram can be calculated by dividing the number of cells A is found in by the total number of cells in the diagram.
Sample space diagrams are particularly useful for calculating probabilities of events with two trials.
For an event A with probability P(A), the expected number of occurrences of A after n trials is given by
This is another way of saying that for every n trials, A will happen an average of P(A)×n times.
Experimental probability (or relative frequency) is found by actually conducting trials and observing outcomes. The relative frequency is calculated by:
While theoretical probability tells us what's expected, experimental probability tells us what's observed.
The complement of an event A, denoted A′, is the event that A does not happen. Since A either happens or it doesn't, then exactly one of A and A′ must happen for each trial:
This expresses the idea that the probability of the entire outcome space is 1.