Concept of Sample space, Event

18/04/2020 0 By indiafreenotes

In probability theory, the sample space (also called sample description space or possibility space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for “universal set”). The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite.

For example, if the experiment is tossing a coin, the sample space is typically the set {head, tail}, commonly written {H, T}. For tossing two coins, the corresponding sample space would be {(head,head), (head,tail), (tail,head), (tail,tail)}, commonly written {HH, HT, TH, TT}. If the sample space is unordered, it becomes {{head,head}, {head,tail}, {tail,tail}}.

For tossing a single six-sided die, the typical sample space is {1, 2, 3, 4, 5, 6} (in which the result of interest is the number of pips facing up).

A subset of the sample space is an event, denoted by E. Referring to the experiment of tossing the coin, the possible events include E={H} and E={T}.

A well-defined sample space is one of three basic elements in a probabilistic model (a probability space); the other two are a well-defined set of possible events (a sigma-algebra) and a probability assigned to each event (a probability measure function).

Another way to look as a sample space is visually. The sample space is typically represented by a rectangle, and the outcomes of the sample space denoted by points within the rectangle. The events are represented by ovals, and the points enclosed within the oval make up the event.

Equally likely outcomes

Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely. For any sample space with N equally likely outcomes, each outcome is assigned the probability 1/N. However, there are experiments that are not easily described by a sample space of equally likely outcomes for example, if one were to toss a thumb tack many times and observe whether it landed with its point upward or downward, there is no symmetry to suggest that the two outcomes should be equally likely.

Though most random phenomena do not have equally likely outcomes, it can be helpful to define a sample space in such a way that outcomes are at least approximately equally likely, since this condition significantly simplifies the computation of probabilities for events within the sample space. If each individual outcome occurs with the same probability, then the probability of any event becomes simply:

P ( e v e n t ) = number of outcomes in event/ number of outcomes in sample space

For example, if two dice are thrown to generate two uniformly distributed integers, D1 and D2, each in the range [1…6], the 36 ordered pairs (D1, D2) constitute a sample space of equally likely events. In this case, the above formula applies, such that the probability of a certain sum, say D1 + D2 = 5 is easily shown to be 4/36, since 4 of the 36 outcomes produce 5 as a sum. On the other hand, the sample space of the 11 possible sums, {2, …,12} are not equally likely outcomes, so the formula would give an incorrect result (1/11).

Another example is having four pens in a bag. One pen is red, one is green, one is blue, and one is purple. Each pen has the same chance of being taken out of the bag.

The sample space S= {red, green, blue, purple}, consists of equally likely events. Here, P(red)=P(blue)=P(green)=P(purple)=1/4.

Sample event

A sample event refers to subset pertaining to sample space that comprises of sample space, empty set (A event that is impossible and possesses zero probability) and singleton set (it is also known as an elementary event). Other event stands to be proper subset pertaining to the sample space which generally comprises of different elements.
Event under the probability theory refers to set of outcome ascertained out of the experiment to which the assigning of probability is done. One outcome might be considered as element pertaining to multiple events that are different from one another. A complimentary event is defined by the event which comprises of event that is not happening and together it defines Bernoulli trial.

Sample event basically refers to something that happens or occurs. For instance flipping of count stands to be an event. Passing by the bench in the park while walking is also counted as an event. There are 1 or more outcomes associated with every event. For example, coin flipping stands to be an event, however, attaining a tail stands to be the outcome of such event. Walking in park stands to be an event, however identifying the friend in park stands to be the outcome associated with the event.

Thus it can be said event stands to be subset pertaining to sample space that comprises of sample space, none or all of the outcome. The event will be considered as a simple event in case it has only 1 sample point. If there are 2 or more sample points underlying an event the same will be considered as compound events. The event will be known as null space in case there is no sample point underlying an event.

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. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event defines a complementary event, namely the complementary set (the event not occurring), and together these define a Bernoulli trial: did the event occur or not?

Typically, when the sample space is finite, any subset of the sample space is an event (i.e. 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 event.

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:

  • “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 (e.g. {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} of an event A is the following formula: