Mathematics
Events (Probability)
In probability theory, an event is a specific outcome or a set of outcomes of a random experiment. It is a subset of the sample space and can be described using set notation. Events are used to calculate the probability of certain outcomes occurring and are fundamental to understanding probability distributions and making predictions in various fields.
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10 Key excerpts on "Events (Probability)"
eBook - PDFClassical and Quantum Information Theory
An Introduction for the Telecom Scientist
- Emmanuel Desurvire(Author)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
Thus, future events, as we may expect them to come out, are well defined facts associated with some degree of likelihood. If we are amidst the Sahara desert or in Paris on a day in November, then rain as an event is associated with a very low or a very high likelihood, respectively. Yet, that day precisely it may rain in the desert or it may shine in Paris, against all preconceived certainties. To make things even more complex (and for that matter, to make life exciting), a few other events may occur, which weren’t included in any of our predictions. Within a given environment of causes and effects, one can make a list of all possible events. The set of events is referred to as an event space (also called sample space). The event space includes anything that can possibly happen. 1 In the case of a sports match between opposing two teams, A and B, for instance, the basic event space is the four-element set: S = team A wins team A loses a draw game canceled , (1.1) with it being implicit that if team A wins, then team B loses, and the reverse. We can then say that the events “team A wins” and “team B loses” are strictly equivalent, and need not be listed twice in the event space. People may take bets as to which team is likely to win (not without some local or affective bias). There may be a draw, or the game may be canceled because of a storm or an earthquake, in that order of likelihood. This pretty much closes the event space. When considering a trial or an experiment, events are referred to as outcomes. An experiment may consist of picking up a card from a 32-card deck. One out of the 32 possible outcomes is the card being the Queen of Hearts. The event space associated 1 In any environment, the list of possible events is generally infinite. One may then conceive of the event space as a limited set of well defined events which encompass all known possibilities at the time of the inventory.
eBook - PDF- Kathleen Subrahmaniam(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
. . , k determine the probability model for a random experiment. From the probability model we can obtain the probability of any event asso-ciated with the experiment. DEFINITION 2.4 The probability of any event E is the sum of the prob-abilities of the simple events which constitute the event E. Two somewhat special cases arise: the entire sample space and the impossible event. Since all the possible outcomes of an experiment must be enumerated in the sample space, Pr(S) = 1. This would be translated as som e event in S must occu r,n which seems very reasonable from the defi-nition of S. If any event is not a possible outcome of the experiment, then it has no corresponding sample points in S. We will call this event an im pos-sible event and its probability is obviously zero. 12 Chapter 2 Referring to our penny-nickel experiment, let us develop an appropri-ate probability model and calculate the probabilities corresponding to the events U, V, W, X, Y and Z. It would seem reasonable, and it can be veri-fied by experimentation, that each outcome is equally likely. Assigning probability 1/4 to each point in S 2 and using Definition 2.4, we find that Pr(U) = 1/2, Pr(V) = 1/4, Pr(W) = 3/4, Pr(X) = 1/2, Pr(Y) = 1/2 and Pr(Z) = 1/4. 2.3 COMBINING EVENTS Since sets and events are analogous, we will now discuss how events, like sets, may be combined. In combining events we are faced with the problem of translating words into logical expressions» In everyday usage, expres-sions of the form A or B” may be interpreted in two different ways: 1. Exclusive A or B but not both 2. Inclusive A or B or both In the following discussion we shall restrict ourselves to the inclusive form . Simultaneous membership in two sets A and B is expressed in words by the terminology and. DEFINITION 2.6 The intersection of the events A and B in S is the set of all points belonging to A and to B» A and B = AB = A fl B = {x |x e A and x < e B }.
eBook - PDF- Howard G Tucker(Author)
- 1998(Publication Date)
- World Scientific(Publisher)
Chapter 1 Events and Probability 1.1 Introduction to Probability The notion of the probability of an event may be approached by at least three methods. One method, perhaps the first historically, is to repeat an experiment or game (in which a certain event might or might not occur) many times under identical conditions and compute the relative frequency with which the event occurs. This means: divide the total number of times that the specific event occurs by the total number of times the experiment is performed or the game is played. This ratio is called the relative frequency and is really only an approximation of what would be considered as the probability of the event. For example, if one tosses a penny 25 times, and if it comes up heads exactly 13 times, then we would estimate the probability that this particular coin will come up heads when tossed is 13/25 or 0.52. Although this method of arriving at the notion of probability is the most primitive and un-sophisticated, it is the most meaningful to the practical individual, in particular, to the working scientist and engineer who have to apply the results of probability theory to real-life situations. Accordingly, what-ever results one obtains in the theory of probability and statistics, one should be able to interpret them in terms of relative frequency. A sec-ond approach to the notion of probability is from an axiomatic point of view. That is, a minimal list of axioms is set down which assumes cer-tain properties of probabilities. From this minimal set of assumptions 1 2 CHAPTER 1. EVENTS AND PROBABILITY the further properties of probabiUty are deduced and applied. A third approach to the notion of probability is limited in applica-tion but is sufficient for our study of sample surveys. This approach is that of probabiUty in the equaUy likely case. Let us consider some game or experiment which, when played or performed, has among its possible outcomes a certain event E.
eBook - PDF- L. Z. Rumshiskii(Author)
- 2016(Publication Date)
- Pergamon(Publisher)
CHAPTER I EVENTS AND PROBABILITIES § 1. E V E N T S . R E L A T I V E F R E Q U E N C Y A N D P R O B A B I L I T Y Let us suppose that we perform an experiment. (A performance of the experiment will be called a trial.) Any possible outcome of a performance of this experiment will be called an event. We will suppose that we may perform the experiment an infinite number of times. EXAMPLE . Trial-the throw of a die: events-the occurrence of a 6; the occurrence of an even number of points. EXAMPLE . Trial-the weighing of an object in an analytical balance: event-the error of measurement (i.e. the difference be-tween the result of the weighing and the true weight of the body) does not exceed a previously given number. If in η trials a given event occurs m times, the relative frequency of the event is the ratio mjn. Experience shows that for repeated trials this relative frequency possesses a definite stability: if, for example, in a large series of η trials the relative frequency is mjn = 0*2, then in another long series of ή trials the relative frequency m'n' will be near 0*2. Thus in different sufficiently long series of trials the relative frequencies of an event appear to be grouped near some constant number (different numbers, of course, for different events). For example, rf a die is a perfect cube, and made of a homogeneous material (an unbiased die) then the numbers 1,2, 3, 4, 5 and 6 will each appear with relative frequency near to . Because of the nature of these experiments, the relative fre-quencies of the events are stable. For example, in the throwing of an unbiased die the approximate equality of the relative fre-quencies with which the six faces appear is explained by its sym-metry, giving the same possibility of occurrence to each number from 1 to 6. ι 2 ELEMENTS OF P R O B A B I L I T Y THEORY Thus we assign to an event a number called the probability of the event.
eBook - PDFFarewell To Entropy, A: Statistical Thermodynamics Based On Information
Statistical Thermodynamics Based on Information
- Arieh Ben-naim(Author)
- 2008(Publication Date)
- World Scientific(Publisher)
It is not only extremely useful but is also an essential tool in all the sciences and beyond. In the axiomatic structure of the theory of probability, the prob-abilities are said to be assigned to each event. These probabilities must subscribe to the three conditions a, b and c . The theory does not define probability, nor provide a method of calculating or mea-suring these probabilities. In fact, there is no way of calculating 3 We exclude the possibility of hitting a specific point or a line — these have zero probability. We also assume that we throw the dart at random , not aiming at any particular area. Elements of Probability Theory 43 probabilities for any general event. It is still a quantity that mea-sures our degree or extent of belief of the occurrence of certain events. As such, it is a highly subjective quantity. However, for some simple experiments, say tossing a coin or throwing a die, we have some very useful methods of calculating probabilities. They have their limitations and they apply to “ideal” cases, yet these probabilities turn out to be extremely useful. What is more impor-tant, since these are based on common sense reasoning, we should all agree that these are the “correct” probabilities, i.e., these prob-abilities turn from subjective quantities into objective quantities. 4 They “belong” to the events as much as mass belongs to a piece of matter. We shall describe two very useful “definitions” that have been suggested and commonly used for this concept. 2.3 The Classical Definition This is sometimes referred to as the a priori definition. 5 Let N ( total ) be the total number of outcomes of a specified experiment, e.g., for throwing a die N ( total ) is six, i.e., the six outcomes (or six elementary events) of this experiment. We denote by N ( event ), the number of outcomes (i.e., elementary events) that are included in the event in which we are interested.
- Carl-Louis Sandblom(Author)
- 2019(Publication Date)
- De Gruyter(Publisher)
42 4. Probability experiment is the set of all possible events that may occur. Exactly one of the events from the event space will occur. In the toss a coin experiment the event space will be {head, tail}. The elements of the event space are sometimes also referred to as simple or elementary events as opposed to composite events, which are combinations of simple events. Example 4.1 Consider the random experiment toss a die and record the number of dots showing. The event space is {1,2,3,4,5,6}. {3} is an elementary event, but {odd value} is the composite event (or event set ) {1, 3, 5}. Suppose that we toss a coin three times and record the outcome for each toss. By (T, H, H) we mean the event: tail in the first toss and head in the second and third. This is not the same event as (H, H, T): head in the first and second tosses and tail in the third. There are 8 simple events for the random experi-ment toss a coin three times, because there are two possibilities for the first toss. For each of these, there are two possibilities for the second toss, which gives 2 x 2 = 4 possibilities for the first two tosses. For each of these four there are two possibilities for the third toss, i.e. 2 x 2 x 2 = 8 possibilities for all three tosses. We can illustrate the outcome of a random experiment like the above where there are several stages, by a so-called tree diagram. See Fig. 4/1 on the next page. The tree grows from left to right and branches out at the event forks. Suppose now that a random experiment is repeated a large number of times, n, and that an event, e, occurs n e times (n e ^ n). We define the probability of the event e, P(e), as the value to which — tends, when n becomes very large: n P(e) = lim —. n *i n Remark: We can not be absolutely sure that — will tend to some number as n n grows, but all empirical evidence tells us that it will. Aswemusthave 0 iS n e ^ n we get 0 ^ — iS 1 and so 0 iS P(e) 1 for every n event.
eBook - PDF- Prem S. Mann(Author)
- 2017(Publication Date)
- Wiley(Publisher)
In other words, these proba- bilities are greater than zero but less than 1.0. A higher probability such as .82 indicates that the event is more likely to occur. On the other hand, an event with a lower probability such as .12 is less likely to occur. Sometime events with very low (.05 or lower) probabilities are also called rare events. 2. The sum of the probabilities of all simple events (or final outcomes) for an experiment, denoted by ∑ P(E i ), is always 1. Second Property of Probability For an experiment with outcomes E 1 , E 2 , E 3 , . . . , ∑ P( E i ) = P( E 1 ) + P( E 2 ) + P( E 3 ) + . . . = 1.0 For example, if you buy a lottery ticket, you may either win or lose. The probabilities of these two events must add to 1.0, that is: P(you will win) + P(you will lose) = 1.0 Similarly, for the experiment of one toss of a coin, P(Head) + P(Tail) = 1.0 For the experiment of two tosses of a coin, P( HH) + P( HT) + P( TH) + P( TT) = 1.0 For one game of football by a professional team, P(win) + P(loss) + P(tie) = 1.0 4.2.2 Three Conceptual Approaches to Probability How do we assign probabilities to events? For example, we may say that the probability of obtaining a head in one toss of a coin is .50, or that the probability that a randomly selected family owns a home is .68, or that the Los Angeles Dodgers will win the Major League Baseball championship next year is .14. How do we obtain these probabilities? We will learn the proce- dures that are used to obtain such probabilities in this section. There are three conceptual approaches to probability: (1) classical probability, (2) the relative frequency concept of proba- bility, and (3) the subjective probability concept. These three concepts are explained next. Classical Probability Many times, various outcomes for an experiment may have the same probability of occurrence.
eBook - PDF- D.G. Rees(Author)
- 2018(Publication Date)
- Chapman and Hall/CRC(Publisher)
We all use ‘subjective probability’ in forecasting future events, for example, when we try to decide whether it will rain tomorrow, and when we try to assess the reactions of others to our opinions and actions. We may not be quite so calculating as to estimate a probability value, but we may regard future events as being probable, rather than just possible. In subjective assessments of probability we may take into account experimental data from past events, but we are likely to add a dose of subjectivity depending on our personality, our mood, and other factors. 5.8 Probabilities Involving More Than One Event Suppose that we are interested in the probabilities of two possible events, El and E 2. For example, we may wish to know the probability that both events will occur, or perhaps the probability that either or both events will occur. We will refer to these as, respectively, P(El and E 2) and P{E or E or both). In set theory notation these compound events are called the intersec-tion and union of events E and E 2, and their probabilities are written: P(El PI E2) and P(E1 U E2) Probability ■ 53 There are two probability laws which can be used to estimate such probabilities, and these are discussed in Sections 5.9 and 5.10. 5.9 Multiplication Law (The 'and' Law) The general case of the m ultiplication law is PiE, and E 2) = P iE JP iE JE J (5.3) where P(E E1) means the probability that event E will occur, given that event E has already occurred. The vertical line between E and E should be read as ‘given that’ or ‘on the condition that’. P(E El ) is an example of what is called a conditional probability. Probability Example 5.4 If two cards are selected at random, one at a time w ithout replacem ent from a pack of 52 playing cards, what is the probability that both cards will be aces? P(two aces) = P(first card is ace and second card is ace), which is logical.
eBook - PDF- Parimal Mukhopadhyay(Author)
- 2011(Publication Date)
- WSPC(Publisher)
This verifies that the probability of any particular face showing up is 1 6 . Such findings form the basis for empirical or statistical definition of probability. In summary we observe that the experiments conducted under identical conditions a large number of times show a statistical regularity, namely, the relative frequency of an outcome in several sets of sequences of trials is more or less constant, provided each set consists of a large number of trials. The rate of convergence of relative frequencies to this particular value increases rapidly as the number of trials increases. This constant value may be taken as the probability of the outcome. The basic requirements (assumptions) for this definition is that the experiments must be conducted under identical conditions and the number of trials must be large. 42 The Classical Approach Definition 2.8.1 : Let f n ( A ) be the number of times in which an event A , the outcome of an experiment, occurs in a series of n repetitions of the trial conducted under identical conditions. The relative frequency of A is r n ( A ) = f n ( A ) /n . The probability of the event A is defined as P ( A ) = lim n →∞ f n ( A ) n provided the limit exists and is unique. Note that even if the conditions of the experiment are such that the el-ementary events are not equally likely, the probability can be defined in the statistical sense, though, however, it remains undefined in the classical sense. 2.9 Geometric Probability Geometric approach to the calculation of probabilities is employed when the sample space Ω includes an uncountable set of elementary events ω , none of which is more likely to occur than the other. Suppose, as in figure 2.7.1, the sample space Ω is a domain in a plane and the elementary events ω are points within Ω. If an event A is represented by a region A within Ω, so that all ω belonging to the region A are favorable to the event A , then the probability of A is P ( A ) = area of subdomain A area of domain Ω .
eBook - PDF- Prem S. Mann(Author)
- 2016(Publication Date)
- Wiley(Publisher)
First Property of Probability 0 ≤ P( E i ) ≤ 1 0 ≤ P( A) ≤ 1 An event that cannot occur has zero probability and is called an impossible (or null) event. An event that is certain to occur has a probability equal to 1 and is called a sure (or certain) event. In the following examples, the first event is an impossible event and the second one is a sure event. P(a tossed coin will stand on its edge) = 0 P(a child born today will eventually die) = 1.0 There are very few events in real life that have probability equal to either zero or 1.0. Most of the events in real life have probabilities that are between zero and 1.0. In other words, these proba- bilities are greater than zero but less than 1.0. A higher probability such as .82 indicates that the event is more likely to occur. On the other hand, an event with a lower probability such as .12 is less likely to occur. Sometime events with very low (.05 or lower) probabilities are also called rare events. 2. The sum of the probabilities of all simple events (or final outcomes) for an experiment, denoted by ∑ P(E i ), is always 1. Second Property of Probability For an experiment with outcomes E 1 , E 2 , E 3 , . . . , ∑ P( E i ) = P( E 1 ) + P( E 2 ) + P( E 3 ) + . . . = 1.0 For example, if you buy a lottery ticket, you may either win or lose. The probabilities of these two events must add to 1.0, that is: P(you will win) + P(you will lose) = 1.0 Similarly, for the experiment of one toss of a coin, P(Head) + P(Tail) = 1.0 For the experiment of two tosses of a coin, P( HH) + P( HT) + P( TH) + P( TT) = 1.0 For one game of football by a professional team, P(win) + P(loss) + P(tie) = 1.0
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