# is probability defined on outcomes or events (or both)?

studying myself and trying to understand probability foundations, and I find that even books do not agree, or probably I do not understand.

What I have learned:

Outcome: what can happen in an experiment. I think these are exclusive, for any experiment only one outcome not several. I think it also called "elementary event". Sample space is the set of all outcome. Example: throw the die, the outcome is 1 or 2 or ... or 6.

Event: a grouping or function(?) on the outcome. Example: the outcome is even.

Random Variable: a mapping from ___ to real numbers.

Probability: a mapping from ___ to probability, that acts like a measure (additive, ...).

The question is about the empty parts ___ above, and specifically for probability (since we can only ask one question each post).

Is probability a map from outcomes, or events? I have looked in different books and get more confused.

Book1 suggests is "outcomes": "All elementary events for a sample space. The probability of a random event is a measure on this set."

Book2: "probabilities are assigned to events"

– whuber
Jan 26, 2020 at 21:51
• page is too advanced for a beginner. I think is possible to answer the question more directly, such as the answer from AP below. Jan 27, 2020 at 18:01
• Does everyone agree the answer from AP is correct? Jan 27, 2020 at 18:02

Outcomes are indeed a word used for the "elementary events". So they are only a subset of the possible "events" to which we can assign probabilities.

Standard example: let a six-size fair die thrown once. The "outcomes/elementary events" are $$(1,2,3,4,5,6)$$.

But an "event" is also, "the die came up even". This is not an elementary event, but we want and can assign probabilities also to it.

Another non-elementary event is "The value of the die is less than 3". Etc.

So probabilities are assigned to events, that include also the elementary events.

A probability is a mapping from the collection of events to [0,1]. Eg, for a dice the event "even outcome" is 1/2. The event "1" takes probability 1/6. The collection of events consists of all subsets of outcomes (called the power set).

A random variable is a mapping from some background space to the real numbers. Eg, for a dice it takes the numbers 1,2,3,4,5,6.

• The term "background space" is new. Does it mean the same as "outcome"? Jan 27, 2020 at 17:58

the simplified, mathematical ituition of a random variable is that it is a function from a measure space into the real numbers.

A measure space is a space with a measure.

And measure is necessary to quantify the probability of the events (and event is a possible outcome of an experiment) in that space.

Example: You flip a coin. Measure Space has two events: Head and tails and it holds the probabilities of those events: eg. 0.4 and 0.6

Now, you dont want a space with heads and tails or other weird things. Therefore the mapping from this weird space with the measure into the real numbers.

Plus: the random variable transmits the information about the event probabilities into the real numbers.

So that flipping a coin becomes a bernoulli experiment with 0-1 outcomes

• The measure space used to describe a coin flip has four events, not two!
– whuber
Jan 26, 2020 at 21:50
• A measure space is a triplet, that includes the "sample space" (elementary events), a sigma algebra based on this sample space that in order to be a sigma algebra, must include the null set and the sample space itself as a whole, and the measure. Since probabilities are assigned to the events in the sigma algebra, we usually count the events in the sigma-algebra, not in the sample space, when communicating informally about the matter. Jan 27, 2020 at 17:05
• Now even more confused about RV. The books I read seem to say it is a function from ___ to reals, where ___ seems to be either outcomes or events. zulop's answer might be the same as "events', but ... it seems instead it is a function from probabilities to numbers? if "probability = measure". Jan 29, 2020 at 5:47
• And I thought that probability comes into RV from the cumulative distribution function ordering: integrate along the real line, gives the cdf, the derivative of that is the probability density, which strangely has nothing to do with the measure on events? Jan 29, 2020 at 5:49