What is a single event in math?

Life is full of random events!

You need to get a "feel" for them to be a smart and successful person.

The toss of a coin, throw of a dice and lottery draws are all examples of random events.

Events

When we say "Event" we mean one (or more) outcomes.

• Getting a Tail when tossing a coin is an event
• Rolling a "5" is an event.

An event can include several outcomes:

• Choosing a "King" from a deck of cards (any of the 4 Kings) is also an event
• Rolling an "even number" (2, 4 or 6) is an event

Events can be:

• Independent (each event is not affected by other events),
• Dependent (also called "Conditional", where an event is affected by other events)
• Mutually Exclusive (events can't happen at the same time)

Let's look at each of those types.

Independent Events

Events can be "Independent", meaning each event is not affected by any other events.

This is an important idea! A coin does not "know" that it came up heads before ... each toss of a coin is a perfect isolated thing.

Example: You toss a coin three times and it comes up "Heads" each time ... what is the chance that the next toss will also be a "Head"?

The chance is simply 1/2, or 50%, just like ANY OTHER toss of the coin.

What it did in the past will not affect the current toss!

Some people think "it is overdue for a Tail", but really truly the next toss of the coin is totally independent of any previous tosses.

Saying "a Tail is due", or "just one more go, my luck is due" is called The Gambler's Fallacy

Learn more at Independent Events.

Dependent Events

But some events can be "dependent" ... which means they can be affected by previous events.

After taking one card from the deck there are less cards available, so the probabilities change!

Let's look at the chances of getting a King.

For the 1st card the chance of drawing a King is 4 out of 52

But for the 2nd card:

• If the 1st card was a King, then the 2nd card is less likely to be a King, as only 3 of the 51 cards left are Kings.
• If the 1st card was not a King, then the 2nd card is slightly more likely to be a King, as 4 of the 51 cards left are King.

This is because we are removing cards from the deck.

Replacement: When we put each card back after drawing it the chances don't change, as the events are independent.

Without Replacement: The chances will change, and the events are dependent.

You can learn more at Dependent Events: Conditional Probability

Tree Diagrams

When we have Dependent Events it helps to make a "Tree Diagram"

You are off to soccer, and love being the Goalkeeper, but that depends who is the Coach today:

• with Coach Sam your probability of being Goalkeeper is 0.5
• with Coach Alex your probability of being Goalkeeper is 0.3

Sam is Coach more often ... about 6 of every 10 games (a probability of 0.6).

Let's build the Tree Diagram!

Start with the Coaches. We know 0.6 for Sam, so it must be 0.4 for Alex (the probabilities must add to 1):

Then fill out the branches for Sam (0.5 Yes and 0.5 No), and then for Alex (0.3 Yes and 0.7 No):

Now it is neatly laid out we can calculate probabilities (read more at Tree Diagrams).

Mutually Exclusive

Mutually Exclusive means we can't get both events at the same time.

It is either one or the other, but not both

Examples:

• Turning left or right are Mutually Exclusive (you can't do both at the same time)
• Heads and Tails are Mutually Exclusive
• Kings and Aces are Mutually Exclusive

What isn't Mutually Exclusive

• Kings and Hearts are not Mutually Exclusive, because we can have a King of Hearts!

Like here:

Aces and Kings are Mutually Exclusive (can't be both)		Hearts and Kings are not Mutually Exclusive (can be both)

Read more at Mutually Exclusive Events

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In statistics and probability theory, set of outcomes to which a probability is assigned

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Part of a series on statistics
Probability theory
Probability Axioms Determinism System Indeterminism Randomness
Probability space Sample space Event Collectively exhaustive events Elementary event Mutual exclusivity Outcome Singleton Experiment Bernoulli trial Probability distribution Bernoulli distribution Binomial distribution Normal distribution Probability measure Random variable Bernoulli process Continuous or discrete Expected value Markov chain Observed value Random walk Stochastic process
Complementary event Joint probability Marginal probability Conditional probability
Independence Conditional independence Law of total probability Law of large numbers Bayes' theorem Boole's inequality
Venn diagram Tree diagram
v t e

In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned.[1] A single outcome may be an element of many different events,[2] and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes.[3] An event consisting of only a single outcome is called an elementary event or an atomic event; that is, it is a singleton set. An event S {\displaystyle S}

x {\displaystyle x}

x ∈ S {\displaystyle x\in S}

Typically, when the sample space is finite, any subset of the sample space is an event (that is, all elements of the power set of the sample space are defined as events). However, this approach does not work well in cases where the sample space is uncountably infinite. So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see Events in probability spaces, below).

A simple example

If we assemble a deck of 52 playing cards with no jokers, and draw a single card from the deck, then the sample space is a 52-element set, as each card is a possible outcome. An event, however, is any subset of the sample space, including any singleton set (an elementary event), the empty set (an impossible event, with probability zero) and the sample space itself (a certain event, with probability one). Other events are proper subsets of the sample space that contain multiple elements. So, for example, potential events include:

B {\displaystyle B}

A {\displaystyle A}

• "Red and black at the same time without being a joker" (0 elements),
• "The 5 of Hearts" (1 element),
• "A King" (4 elements),
• "A Face card" (12 elements),
• "A Spade" (13 elements),
• "A Face card or a red suit" (32 elements),
• "A card" (52 elements).

Since all events are sets, they are usually written as sets (for example, {1, 2, 3}), and represented graphically using Venn diagrams. In the situation where each outcome in the sample space Ω is equally likely, the probability P {\displaystyle P}

P ( A ) = | A | | Ω |   ( alternatively:   Pr ( A ) = | A | | Ω | ) {\displaystyle \mathrm {P} (A)={\frac {|A|}{|\Omega |}}\,\ \left({\text{alternatively:}}\ \Pr(A)={\frac {|A|}{|\Omega |}}\right)}

Events in probability spaces

Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standard probability distributions, such as the normal distribution, the sample space is the set of real numbers or some subset of the real numbers. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers 'badly behaved' sets, such as those that are nonmeasurable. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as joint and conditional probabilities, to work, it is necessary to use a σ-algebra, that is, a family closed under complementation and countable unions of its members. The most natural choice of σ-algebra is the Borel measurable set derived from unions and intersections of intervals. However, the larger class of Lebesgue measurable sets proves more useful in practice.

In the general measure-theoretic description of probability spaces, an event may be defined as an element of a selected 𝜎-algebra of subsets of the sample space. Under this definition, any subset of the sample space that is not an element of the 𝜎-algebra is not an event, and does not have a probability. With a reasonable specification of the probability space, however, all events of interest are elements of the 𝜎-algebra.

A note on notation

Even though events are subsets of some sample space Ω , {\displaystyle \Omega ,}

X {\displaystyle X}

{ ω ∈ Ω ∣ u < X ( ω ) ≤ v } {\displaystyle \{\omega \in \Omega \mid u<X(\omega )\leq v\}\,}

u < X ≤ v . {\displaystyle u<X\leq v\,.}

Pr ( u < X ≤ v ) = F ( v ) − F ( u ) . {\displaystyle \Pr(u<X\leq v)=F(v)-F(u)\,.}

u < X ≤ v {\displaystyle u<X\leq v}

ω ∈ X − 1 ( ( u , v ] ) {\displaystyle \omega \in X^{-1}((u,v])}

u < X ( ω ) ≤ v . {\displaystyle u<X(\omega )\leq v.}

See also

• Atom (measure theory)
• Complementary event – Opposite of a probability event
• Elementary event
• Independent event
• Outcome (probability)
• Pairwise independent events

Notes

1. ^ Leon-Garcia, Alberto (2008). Probability, statistics and random processes for electrical engineering. Upper Saddle River, NJ: Pearson. ISBN 9780131471221.
2. ^ Pfeiffer, Paul E. (1978). Concepts of probability theory. Dover Publications. p. 18. ISBN 978-0-486-63677-1.
3. ^ Foerster, Paul A. (2006). Algebra and trigonometry: Functions and applications, Teacher's edition (Classics ed.). Upper Saddle River, NJ: Prentice Hall. p. 634. ISBN 0-13-165711-9.

External links

Wikimedia Commons has media related to Event (probability theory).

• "Random event", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Formal definition in the Mizar system.

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