# 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"

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"? – stochasticthoughtprocess Jan 27 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 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. – Alecos Papadopoulos Jan 27 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". – stochasticthoughtprocess Jan 29 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? – stochasticthoughtprocess Jan 29 at 5:49