Solved – Are the terms “event” and “outcome” synonymous
probabilityterminology
An outcome is a result of a random experiment and an event is a single result of an experiment.
Are the terms "event" and "outcome" synonymous?
Best Answer
Outcome and event are not synonymous.
Yes, an outcome is the result of a random experiment, like a rolling a die has six possible outcomes (say). However, an "event" is a set of outcomes to which a probability is assigned. One possible event is "rolling a number less than 3". See the Wikipedia page for probability theory and probability space for better descriptions.
Let the experiment be given by $ \DeclareMathOperator{\P}{\mathbb{P}} \DeclareMathOperator{\E}{\mathbb{E}} (\mathbb{X},\mathbb{B}, \P)$ where $\mathbb{X}$ is the sample space, $\mathbb{B}$ is the set of all events (subsets of $\mathbb{X}$ which we assign a probability) and $\P$ is the probability measure. Points of $\mathbb{X}$ are denoted $\omega$, and are the "elementary events" (or "outcomes"). Random variables on this experiment are functions $f \colon \mathbb{X}\mapsto \mathbb{R}$ and are written like $f(\omega)$, meaning that their value are determined by the elementary outcome $\omega$.
Corresponding to the event $A$ is the indicator random variable
$$
I_A(\omega) = \begin{cases} 1 ~\text{if $A$ occurs, that is, $\omega\in A$.} \\
0 ~\text{if $A$ do not occur, that is $\omega \not\in A$.} \end{cases}
$$
In this sense, events can be embedded as a subset of the set of all random variables defined for this experimental setup. Then the probability of $A$ occurring can be written as an expectation
$$
\P(A) = \E I_A.
$$
To the additional question in comments: If $A$ and $B$ are independent (as events), then $I_A$ and $I_B$ are independent (as random variables). "can we say that I_A=1 and I_B=1 are independent?" Well, $I_A=1$ is simply the event $A$, so I think you can answer now!
Best Answer
Outcome and event are not synonymous.
Yes, an outcome is the result of a random experiment, like a rolling a die has six possible outcomes (say). However, an "event" is a set of outcomes to which a probability is assigned. One possible event is "rolling a number less than 3". See the Wikipedia page for probability theory and probability space for better descriptions.