Misplaced Pages

#409590

82-508: The Drake equation is a probabilistic argument used to estimate the number of active, communicative extraterrestrial civilizations in the Milky Way Galaxy . The equation was formulated in 1961 by Frank Drake , not for purposes of quantifying the number of civilizations, but as a way to stimulate scientific dialogue at the first scientific meeting on the search for extraterrestrial intelligence (SETI). The equation summarizes

164-714: A counting measure over the set of all possible outcomes. Densities for absolutely continuous distributions are usually defined as this derivative with respect to the Lebesgue measure . If a theorem can be proved in this general setting, it holds for both discrete and continuous distributions as well as others; separate proofs are not required for discrete and continuous distributions. Certain random variables occur very often in probability theory because they well describe many natural or physical processes. Their distributions, therefore, have gained special importance in probability theory. Some fundamental discrete distributions are

246-469: A measure P {\displaystyle P\,} defined on F {\displaystyle {\mathcal {F}}\,} is called a probability measure if P ( Ω ) = 1. {\displaystyle P(\Omega )=1.\,} If F {\displaystyle {\mathcal {F}}\,} is the Borel σ-algebra on the set of real numbers, then there

328-467: A sequence of independent and identically distributed random variables X k {\displaystyle X_{k}} converges towards their common expectation (expected value) μ {\displaystyle \mu } , provided that the expectation of | X k | {\displaystyle |X_{k}|} is finite. It is in the different forms of convergence of random variables that separates

410-511: A wavelength of 21 cm (1,420.4  MHz ). This is the wavelength of radio emission by neutral hydrogen , the most common element in the universe, and they reasoned that other intelligences might see this as a logical landmark in the radio spectrum . Two months later, Harvard University astronomy professor Harlow Shapley speculated on the number of inhabited planets in the universe, saying "The universe has 10 million, million, million suns (10 followed by 18 zeros) similar to our own. One in

492-502: A book on the subject in 1657. In the 19th century, what is considered the classical definition of probability was completed by Pierre Laplace . Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial . Eventually, analytical considerations compelled the incorporation of continuous variables into the theory. This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov . Kolmogorov combined

574-491: A civilization has developed enough, it might overcome all threats to its survival. It will then last for an indefinite period of time, making the value for L potentially billions of years. If this is the case, then he proposes that the Milky Way Galaxy may have been steadily accumulating advanced civilizations since it formed. He proposes that the last factor L be replaced with f IC · T , where f IC

656-548: A common origin. If abiogenesis were more common it would be speculated to have occurred more than once on the Earth. Scientists have searched for this by looking for bacteria that are unrelated to other life on Earth, but none have been found yet. It is also possible that life arose more than once, but that other branches were out-competed, or died in mass extinctions, or were lost in other ways. Biochemists Francis Crick and Leslie Orgel laid special emphasis on this uncertainty: "At

738-636: A continuous sample space. Classical definition : The classical definition breaks down when confronted with the continuous case. See Bertrand's paradox . Modern definition : If the sample space of a random variable X is the set of real numbers ( R {\displaystyle \mathbb {R} } ) or a subset thereof, then a function called the cumulative distribution function ( CDF ) F {\displaystyle F\,} exists, defined by F ( x ) = P ( X ≤ x ) {\displaystyle F(x)=P(X\leq x)\,} . That is, F ( x ) returns

820-460: A million has planets around it. Only one in a million million has the right combination of chemicals, temperature, water, days and nights to support planetary life as we know it. This calculation arrives at the estimated figure of 100 million worlds where life has been forged by evolution." Seven months after Cocconi and Morrison published their article, Drake began searching for extraterrestrial intelligence in an experiment called Project Ozma . It

902-508: A mix, for example, the Cantor distribution has no positive probability for any single point, neither does it have a density. The modern approach to probability theory solves these problems using measure theory to define the probability space : Given any set Ω {\displaystyle \Omega \,} (also called sample space ) and a σ-algebra F {\displaystyle {\mathcal {F}}\,} on it,

SECTION 10

#1732780054410

984-472: A more advanced one, its longevity could increase because it can learn from the experiences of the other. The astronomer Carl Sagan speculated that all of the terms, except for the lifetime of a civilization, are relatively high and the determining factor in whether there are large or small numbers of civilizations in the universe is the civilization lifetime, or in other words, the ability of technological civilizations to avoid self-destruction. In Sagan's case,

1066-411: A planet with plate tectonics , a large moon that creates tidal pools, and moderate axial tilt to generate seasonal variation. Geological evidence from the Earth suggests that f l may be high; life on Earth appears to have begun around the same time as favorable conditions arose, suggesting that abiogenesis may be relatively common once conditions are right. However, this evidence only looks at

1148-410: A probability of 1 (certainty). Their resultant calculation concludes there are more than thirty current technological civilizations in the galaxy (disregarding error bars). This value remains particularly controversial. Those who favor a low value, such as the biologist Ernst Mayr , point out that of the billions of species that have existed on Earth, only one has become intelligent and from this, infer

1230-629: A random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem . As a mathematical foundation for statistics , probability theory is essential to many human activities that involve quantitative analysis of data. Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics or sequential estimation . A great discovery of twentieth-century physics

1312-493: A random value from a normal distribution with probability 1/2. It can still be studied to some extent by considering it to have a PDF of ( δ [ x ] + φ ( x ) ) / 2 {\displaystyle (\delta [x]+\varphi (x))/2} , where δ [ x ] {\displaystyle \delta [x]} is the Dirac delta function . Other distributions may not even be

1394-587: A special long-range unmanned spaceship". In 2020, a paper by scholars at the University of Nottingham proposed an "Astrobiological Copernican" principle, based on the Principle of Mediocrity , and speculated that "intelligent life would form on other [Earth-like] planets like it has on Earth, so within a few billion years life would automatically form as a natural part of evolution". In the authors' framework, f l , f i , and f c are all set to

1476-413: A sufficient time. The discovery of numerous gas giants in close orbit with their stars has introduced doubt that life-supporting planets commonly survive the formation of their stellar systems. So-called hot Jupiters may migrate from distant orbits to near orbits, in the process disrupting the orbits of habitable planets. On the other hand, the variety of star systems that might have habitable zones

1558-461: A tiny value for f i . Likewise, the Rare Earth hypothesis, notwithstanding their low value for n e above, also think a low value for f i dominates the analysis. Those who favor higher values note the generally increasing complexity of life over time, concluding that the appearance of intelligence is almost inevitable, implying an f i approaching 1. Skeptics point out that

1640-453: A value for f l close to 1. While this would raise the number of degrees of freedom from zero to one, there would remain a great deal of uncertainty on any estimate due to the small sample size, and the chance they are not really independent. Countering this argument is that there is no evidence for abiogenesis occurring more than once on the Earth—that is, all terrestrial life stems from

1722-662: Is a Bayesian analysis published in 2020. In the conclusion, the author cautions that this study applies to Earth's conditions. In Bayesian terms, the study favors the formation of intelligence on a planet with identical conditions to Earth but does not do so with high confidence. Planetary scientist Pascal Lee of the SETI Institute proposes that this fraction is very low (0.0002). He based this estimate on how long it took Earth to develop intelligent life (1 million years since Homo erectus evolved, compared to 4.6 billion years since Earth formed). For deliberate communication,

SECTION 20

#1732780054410

1804-488: Is a unique probability measure on F {\displaystyle {\mathcal {F}}\,} for any CDF, and vice versa. The measure corresponding to a CDF is said to be induced by the CDF. This measure coincides with the pmf for discrete variables and PDF for continuous variables, making the measure-theoretic approach free of fallacies. The probability of a set E {\displaystyle E\,} in

1886-456: Is attached, which satisfies the following properties: That is, the probability function f ( x ) lies between zero and one for every value of x in the sample space Ω , and the sum of f ( x ) over all values x in the sample space Ω is equal to 1. An event is defined as any subset E {\displaystyle E\,} of the sample space Ω {\displaystyle \Omega \,} . The probability of

1968-469: Is given by the sum of the probabilities of the events. The probability that any one of the events {1,6}, {3}, or {2,4} will occur is 5/6. This is the same as saying that the probability of event {1,2,3,4,6} is 5/6. This event encompasses the possibility of any number except five being rolled. The mutually exclusive event {5} has a probability of 1/6, and the event {1,2,3,4,5,6} has a probability of 1, that is, absolute certainty. When doing calculations using

2050-433: Is not just limited to solar-type stars and Earth-sized planets. It is now estimated that even tidally locked planets close to red dwarf stars might have habitable zones , although the flaring behavior of these stars might speak against this. The possibility of life on moons of gas giants (such as Jupiter 's moon Europa , or Saturn 's moons Titan and Enceladus ) adds further uncertainty to this figure. The authors of

2132-421: Is the fraction of communicating civilizations that become "immortal" (in the sense that they simply do not die out), and T representing the length of time during which this process has been going on. This has the advantage that T would be a relatively easy-to-discover number, as it would simply be some fraction of the age of the universe. It has also been hypothesized that once a civilization has learned of

2214-853: The Generalized Central Limit Theorem (GCLT). Green Bank, West Virginia Green Bank is a census-designated place in Pocahontas County in West Virginia 's Potomac Highlands inside the Allegheny Mountain Range . Green Bank is located along WV 28 . Green Bank is home to the Green Bank Observatory and is also close to the Snowshoe Mountain ski resort. As of the 2020 census , its population

2296-424: The discrete uniform , Bernoulli , binomial , negative binomial , Poisson and geometric distributions . Important continuous distributions include the continuous uniform , normal , exponential , gamma and beta distributions . In probability theory, there are several notions of convergence for random variables . They are listed below in the order of strength, i.e., any subsequent notion of convergence in

2378-470: The habitable zones of sun-like stars and red dwarf stars within the Milky Way Galaxy . 11 billion of these estimated planets may be orbiting sun-like stars. Since there are about 100 billion stars in the galaxy, this implies f p · n e is roughly 0.4. The nearest planet in the habitable zone is Proxima Centauri b , which is as close as about 4.2 light-years away. The consensus at

2460-699: The identity function . This does not always work. For example, when flipping a coin the two possible outcomes are "heads" and "tails". In this example, the random variable X could assign to the outcome "heads" the number "0" ( X ( heads ) = 0 {\textstyle X({\text{heads}})=0} ) and to the outcome "tails" the number "1" ( X ( tails ) = 1 {\displaystyle X({\text{tails}})=1} ). Discrete probability theory deals with events that occur in countable sample spaces. Examples: Throwing dice , experiments with decks of cards , random walk , and tossing coins . Classical definition : Initially

2542-522: The observable universe . On the other hand, with larger values for each of the parameters above, values of N can be derived that are greater than 1. The following higher values that have been proposed for each of the parameters: Use of these parameters gives: Monte Carlo simulations of estimates of the Drake equation factors based on a stellar and planetary model of the Milky Way have resulted in

Drake equation - Misplaced Pages Continue

2624-403: The rare Earth hypothesis propose a number of additional constraints on habitability for planets, including being in galactic zones with suitably low radiation, high star metallicity, and low enough density to avoid excessive asteroid bombardment. They also propose that it is necessary to have a planetary system with large gas giants which provide bombardment protection without a hot Jupiter ; and

2706-427: The snowball Earth or research into extinction events have raised the possibility that life on Earth is relatively fragile. Research on any past life on Mars is relevant since a discovery that life did form on Mars but ceased to exist might raise the estimate of f l but would indicate that in half the known cases, intelligent life did not develop. Estimates of f i have been affected by discoveries that

2788-886: The weak and the strong law of large numbers It follows from the LLN that if an event of probability p is observed repeatedly during independent experiments, the ratio of the observed frequency of that event to the total number of repetitions converges towards p . For example, if Y 1 , Y 2 , . . . {\displaystyle Y_{1},Y_{2},...\,} are independent Bernoulli random variables taking values 1 with probability p and 0 with probability 1- p , then E ( Y i ) = p {\displaystyle {\textrm {E}}(Y_{i})=p} for all i , so that Y ¯ n {\displaystyle {\bar {Y}}_{n}} converges to p almost surely . The central limit theorem (CLT) explains

2870-412: The 21 cm wavelength of the hydrogen frequency . There is considerable disagreement on the values of these parameters, but the 'educated guesses' used by Drake and his colleagues in 1961 were: Inserting the above minimum numbers into the equation gives a minimum N of 20 (see: Range of results ). Inserting the maximum numbers gives a maximum of 50,000,000. Drake states that given the uncertainties,

2952-556: The 6 have even numbers and each face has the same probability of appearing. Modern definition : The modern definition starts with a finite or countable set called the sample space , which relates to the set of all possible outcomes in classical sense, denoted by Ω {\displaystyle \Omega } . It is then assumed that for each element x ∈ Ω {\displaystyle x\in \Omega \,} , an intrinsic "probability" value f ( x ) {\displaystyle f(x)\,}

3034-557: The Dolphin" (because of Lilly's work on dolphin communication ), and commemorated their first meeting with a plaque at the observatory hall. The Drake equation results in a summary of the factors affecting the likelihood that we might detect radio-communication from intelligent extraterrestrial life. The last three parameters, f i , f c , and L , are not known and are very difficult to estimate, with values ranging over many orders of magnitude (see § Criticism ). Therefore,

3116-432: The Drake equation was a strong motivating factor for his interest in environmental issues and his efforts to warn against the dangers of nuclear warfare . Paleobiologist Olev Vinn suggests that the lifetime of most technological civilizations is brief due to inherited behavior patterns present in all intelligent organisms. These behaviors, incompatible with civilized conditions, inevitably lead to self-destruction soon after

3198-576: The Earth (a single model planet), and contains anthropic bias , as the planet of study was not chosen randomly, but by the living organisms that already inhabit it (ourselves). From a classical hypothesis testing standpoint, without assuming that the underlying distribution of f l is the same for all planets in the Milky Way, there are zero degrees of freedom , permitting no valid estimates to be made. If life (or evidence of past life) were to be found on Mars , Europa , Enceladus or Titan that developed independently from life on Earth it would imply

3280-459: The Earth is a communicating civilization. Another question is what percentage of civilizations in the galaxy are close enough for us to detect, assuming that they send out signals. For example, existing Earth radio telescopes could only detect Earth radio transmissions from roughly a light year away. Michael Shermer estimated L as 420 years, based on the duration of sixty historical Earthly civilizations. Using 28 civilizations more recent than

3362-566: The Green Bank meeting was that n e had a minimum value between 3 and 5. Dutch science journalist Govert Schilling has opined that this is optimistic. Even if planets are in the habitable zone , the number of planets with the right proportion of elements is difficult to estimate. Brad Gibson, Yeshe Fenner, and Charley Lineweaver determined that about 10% of star systems in the Milky Way Galaxy are hospitable to life, by having heavy elements, being far from supernovae and being stable for

Drake equation - Misplaced Pages Continue

3444-541: The Roman Empire, he calculates a figure of 304 years for "modern" civilizations. It could also be argued from Michael Shermer's results that the fall of most of these civilizations was followed by later civilizations that carried on the technologies, so it is doubtful that they are separate civilizations in the context of the Drake equation. In the expanded version, including reappearance number , this lack of specificity in defining single civilizations does not matter for

3526-539: The Solar System's orbit is circular in the galaxy, at such a distance that it remains out of the spiral arms for tens of millions of years (evading radiation from novae ). Also, Earth's large moon may aid the evolution of life by stabilizing the planet's axis of rotation . There has been quantitative work to begin to define f l ⋅ f i {\displaystyle f_{\mathrm {l} }\cdot f_{\mathrm {i} }} . One example

3608-473: The backbone of astrobiology as a science; although speculation is entertained to give context, astrobiology concerns itself primarily with hypotheses that fit firmly into existing scientific theories . Some 50 years of SETI have failed to find anything, even though radio telescopes, receiver techniques, and computational abilities have improved significantly since the early 1960s. SETI efforts since 1961 have conclusively ruled out widespread alien emissions near

3690-410: The derivative gives us the CDF back again, then the random variable X is said to have a probability density function ( PDF ) or simply density f ( x ) = d F ( x ) d x . {\displaystyle f(x)={\frac {dF(x)}{dx}}\,.} For a set E ⊆ R {\displaystyle E\subseteq \mathbb {R} } ,

3772-460: The development of multi-cellular life , and the development of intelligence itself. Within the limits of existing human technology, any practical search for distant intelligent life must necessarily be a search for some manifestation of a distant technology. After about 50 years, the Drake equation is still of seminal importance because it is a 'road map' of what we need to learn in order to solve this fundamental existential question. It also formed

3854-490: The discrete, continuous, a mix of the two, and more. Consider an experiment that can produce a number of outcomes. The set of all outcomes is called the sample space of the experiment. The power set of the sample space (or equivalently, the event space) is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus,

3936-453: The emergence of advanced technologies. An intelligent civilization might not be organic, as some have suggested that artificial general intelligence may replace humanity. As many skeptics have pointed out, the Drake equation can give a very wide range of values, depending on the assumptions, as the values used in portions of the Drake equation are not well established. In particular, the result can be N ≪ 1 , meaning we are likely alone in

4018-476: The equation first appeared in Drake's 1965 paper. In September 1959, physicists Giuseppe Cocconi and Philip Morrison published an article in the journal Nature with the provocative title "Searching for Interstellar Communications". Cocconi and Morrison argued that radio telescopes had become sensitive enough to pick up transmissions that might be broadcast into space by civilizations orbiting other stars. Such messages, they suggested, might be transmitted at

4100-453: The event E {\displaystyle E\,} is defined as So, the probability of the entire sample space is 1, and the probability of the null event is 0. The function f ( x ) {\displaystyle f(x)\,} mapping a point in the sample space to the "probability" value is called a probability mass function abbreviated as pmf . Continuous probability theory deals with events that occur in

4182-427: The event made up of all possible results (in our example, the event {1,2,3,4,5,6}) be assigned a value of one. To qualify as a probability distribution , the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events (events that contain no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), the probability that any of these events occurs

SECTION 50

#1732780054410

4264-401: The foundations of probability theory, but instead emerges from these foundations as a theorem. Since it links theoretically derived probabilities to their actual frequency of occurrence in the real world, the law of large numbers is considered as a pillar in the history of statistical theory and has had widespread influence. The law of large numbers (LLN) states that the sample average of

4346-563: The galaxy, or N ≫ 1 , implying there are many civilizations we might contact. One of the few points of wide agreement is that the presence of humanity implies a probability of intelligence arising of greater than zero. As an example of a low estimate, combining NASA's star formation rates, the rare Earth hypothesis value of f p · n e · f l = 10 , Mayr's view on intelligence arising, Drake's view of communication, and Shermer's estimate of lifetime: gives: i.e., suggesting that we are probably alone in this galaxy, and possibly in

4428-503: The large spread of values in this factor and others make all estimates unreliable. (See Criticism ). In addition, while it appears that life developed soon after the formation of Earth, the Cambrian explosion , in which a large variety of multicellular life forms came into being, occurred a considerable amount of time after the formation of Earth, which suggests the possibility that special conditions were necessary. Some scenarios such as

4510-406: The list implies convergence according to all of the preceding notions. As the names indicate, weak convergence is weaker than strong convergence. In fact, strong convergence implies convergence in probability, and convergence in probability implies weak convergence. The reverse statements are not always true. Common intuition suggests that if a fair coin is tossed many times, then roughly half of

4592-441: The main concepts which scientists must contemplate when considering the question of other radio-communicative life. It is more properly thought of as an approximation than as a serious attempt to determine a precise number. Criticism related to the Drake equation focuses not on the equation itself, but on the fact that the estimated values for several of its factors are highly conjectural, the combined multiplicative effect being that

4674-433: The measure-theoretic treatment of probability is that it unifies the discrete and the continuous cases, and makes the difference a question of which measure is used. Furthermore, it covers distributions that are neither discrete nor continuous nor mixtures of the two. An example of such distributions could be a mix of discrete and continuous distributions—for example, a random variable that is 0 with probability 1/2, and takes

4756-420: The moment we have no means at all of knowing" whether we are "likely to be alone in the galaxy (Universe)" or whether "the galaxy may be pullulating with life of many different forms." As an alternative to abiogenesis on Earth, they proposed the hypothesis of directed panspermia , which states that Earth life began with "microorganisms sent here deliberately by a technological society on another planet, by means of

4838-540: The notion of sample space , introduced by Richard von Mises , and measure theory and presented his axiom system for probability theory in 1933. This became the mostly undisputed axiomatic basis for modern probability theory; but, alternatives exist, such as the adoption of finite rather than countable additivity by Bruno de Finetti . Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The measure theory-based treatment of probability covers

4920-406: The number of civilizations varying by a factor of 100. Probability theory Central subjects in probability theory include discrete and continuous random variables , probability distributions , and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in

5002-624: The number of stars per year, we divide this by the initial mass function (IMF) for stars, where the average new star's mass is about 0.5  M ☉ . This gives a star formation rate of about 1.5–3 stars per year. Analysis of microlensing surveys, in 2012, has found that f p may approach 1—that is, stars are orbited by planets as a rule, rather than the exception; and that there are one or more bound planets per Milky Way star. In November 2013, astronomers reported, based on Kepler space telescope data, that there could be as many as 40 billion Earth-sized planets orbiting in

SECTION 60

#1732780054410

5084-633: The one example we have (the Earth) does not do much explicit communication, though there are some efforts covering only a tiny fraction of the stars that might look for human presence. (See Arecibo message , for example). There is considerable speculation why an extraterrestrial civilization might exist but choose not to communicate. However, deliberate communication is not required, and calculations indicate that current or near-future Earth-level technology might well be detectable to civilizations not too much more advanced than present day humans. By this standard,

5166-605: The original meeting concluded that N ≈ L , and there were probably between 1000 and 100,000,000 planets with civilizations in the Milky Way Galaxy. This section discusses and attempts to list the best current estimates for the parameters of the Drake equation. Calculations in 2010, from NASA and the European Space Agency indicate that the rate of star formation in this Galaxy is about 0.68–1.45  M ☉ of material per year. To get

5248-411: The outcomes of an experiment, it is necessary that all those elementary events have a number assigned to them. This is done using a random variable . A random variable is a function that assigns to each elementary event in the sample space a real number . This function is usually denoted by a capital letter. In the case of a die, the assignment of a number to certain elementary events can be done using

5330-470: The probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability . For example, if the event is "occurrence of an even number when a dice is rolled", the probability is given by 3 6 = 1 2 {\displaystyle {\tfrac {3}{6}}={\tfrac {1}{2}}} , since 3 faces out of

5412-733: The probability of the random variable X being in E {\displaystyle E\,} is In case the PDF exists, this can be written as Whereas the PDF exists only for continuous random variables, the CDF exists for all random variables (including discrete random variables) that take values in R . {\displaystyle \mathbb {R} \,.} These concepts can be generalized for multidimensional cases on R n {\displaystyle \mathbb {R} ^{n}} and other continuous sample spaces. The utility of

5494-450: The probability that X will be less than or equal to x . The CDF necessarily satisfies the following properties. The random variable X {\displaystyle X} is said to have a continuous probability distribution if the corresponding CDF F {\displaystyle F} is continuous. If F {\displaystyle F\,} is absolutely continuous , i.e., its derivative exists and integrating

5576-410: The result, since such a civilization turnover could be described as an increase in the reappearance number rather than increase in L , stating that a civilization reappears in the form of the succeeding cultures. Furthermore, since none could communicate over interstellar space, the method of comparing with historical civilizations could be regarded as invalid. David Grinspoon has argued that once

5658-560: The sequence of random variables converges in distribution to a standard normal random variable. For some classes of random variables, the classic central limit theorem works rather fast, as illustrated in the Berry–Esseen theorem . For example, the distributions with finite first, second, and third moment from the exponential family ; on the other hand, for some random variables of the heavy tail and fat tail variety, it works very slowly or may not work at all: in such cases one may use

5740-400: The subset {1,3,5} is an element of the power set of the sample space of dice rolls. These collections are called events . In this case, {1,3,5} is the event that the die falls on some odd number. If the results that actually occur fall in a given event, that event is said to have occurred. Probability is a way of assigning every "event" a value between zero and one, with the requirement that

5822-558: The theory of stochastic processes . For example, to study Brownian motion , probability is defined on a space of functions. When it is convenient to work with a dominating measure, the Radon-Nikodym theorem is used to define a density as the Radon-Nikodym derivative of the probability distribution of interest with respect to this dominating measure. Discrete densities are usually defined as this derivative with respect to

5904-779: The things you needed to know to predict how hard it's going to be to detect extraterrestrial life. And looking at them it became pretty evident that if you multiplied all these together, you got a number, N, which is the number of detectable civilizations in our galaxy. This was aimed at the radio search, and not to search for primordial or primitive life forms. The ten attendees were conference organizer J. Peter Pearman, Frank Drake, Philip Morrison , businessman and radio amateur Dana Atchley, chemist Melvin Calvin , astronomer Su-Shu Huang , neuroscientist John C. Lilly , inventor Barney Oliver , astronomer Carl Sagan , and radio-astronomer Otto Struve . These participants called themselves "The Order of

5986-407: The time it will turn up heads , and the other half it will turn up tails . Furthermore, the more often the coin is tossed, the more likely it should be that the ratio of the number of heads to the number of tails will approach unity. Modern probability theory provides a formal version of this intuitive idea, known as the law of large numbers . This law is remarkable because it is not assumed in

6068-756: The ubiquitous occurrence of the normal distribution in nature, and this theorem, according to David Williams, "is one of the great results of mathematics." The theorem states that the average of many independent and identically distributed random variables with finite variance tends towards a normal distribution irrespective of the distribution followed by the original random variables. Formally, let X 1 , X 2 , … {\displaystyle X_{1},X_{2},\dots \,} be independent random variables with mean μ {\displaystyle \mu } and variance σ 2 > 0. {\displaystyle \sigma ^{2}>0.\,} Then

6150-569: The uncertainty associated with any derived value is so large that the equation cannot be used to draw firm conclusions. The Drake equation is: N = R ∗ ⋅ f p ⋅ n e ⋅ f l ⋅ f i ⋅ f c ⋅ L {\displaystyle N=R_{*}\cdot f_{\mathrm {p} }\cdot n_{\mathrm {e} }\cdot f_{\mathrm {l} }\cdot f_{\mathrm {i} }\cdot f_{\mathrm {c} }\cdot L} where and This form of

6232-416: The usefulness of the Drake equation is not in the solving, but rather in the contemplation of all the various concepts which scientists must incorporate when considering the question of life elsewhere, and gives the question of life elsewhere a basis for scientific analysis . The equation has helped draw attention to some particular scientific problems related to life in the universe, for example abiogenesis ,

6314-561: The σ-algebra F {\displaystyle {\mathcal {F}}\,} is defined as where the integration is with respect to the measure μ F {\displaystyle \mu _{F}\,} induced by F . {\displaystyle F\,.} Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outside R n {\displaystyle \mathbb {R} ^{n}} , as in

6396-608: Was 141. The community was named for a green riverbank near the original town site. Green Bank is located within the National Radio Quiet Zone , which means that radio transmissions are heavily restricted by law. This policy is enforced by a "radio policeman" who uses specialized equipment to detect signals from unauthorized electronics. Green Bank is home to the Green Bank Telescope , the world's largest fully steerable radio telescope , which

6478-744: Was operated by the National Radio Astronomy Observatory (NRAO) until September 30, 2016. Since October 1, 2016, the Telescope has been operated by the Green Bank Observatory , which is no longer part of the NRAO. It was at the Green Bank Site in 1961 that Frank Drake presented the Drake Equation , which was developed to provide an estimate of the total number of detectable extraterrestrial civilizations in

6560-479: Was the first systematic search for signals from communicative extraterrestrial civilizations. Using the 85 ft (26 m) dish of the National Radio Astronomy Observatory, Green Bank in Green Bank, West Virginia , Drake monitored two nearby Sun-like stars: Epsilon Eridani and Tau Ceti , slowly scanning frequencies close to the 21 cm wavelength for six hours per day from April to July 1960. The project

6642-403: Was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics . The modern mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the " problem of points "). Christiaan Huygens published

6724-468: Was well designed, inexpensive, and simple by today's standards. It detected no signals. Soon thereafter, Drake hosted the first search for extraterrestrial intelligence conference on detecting their radio signals. The meeting was held at the Green Bank facility in 1961. The equation that bears Drake's name arose out of his preparations for the meeting. As I planned the meeting, I realized a few day[s] ahead of time we needed an agenda. And so I wrote down all

#409590