title: Pleiades

text: More recent results using very-long-baseline interferometry (VLBI) (August 2014) and preliminary solutions using Gaia Data Release 1 (September 2016) and Gaia Data Release 2 (August 2018), determine distances of 136.2 ± 1.2 pc, 134 ± 6 pc and 136.2 ± 5.0 pc, respectively. The Gaia Data Release 1 team was cautious about their result and the VLBI authors assert "that the "Hipparcos"-measured distance to the Pleiades cluster is in error".


title: Pleiades

text: For another distance debate see Polaris#Distance, also with a different measurement from "Hipparcos", although this time it suggested a greater distance.


title: Pleiades

text: The cluster core radius is about 8 light-years and tidal radius is about 43 light-years. The cluster contains over 1,000 statistically confirmed members, a figure that excludes an unresolved likely further number of binary stars. Its light is dominated by young, hot blue stars, up to 14 of which can be seen with the naked eye depending on local observing conditions and visual acuity of the observer. The arrangement of the brightest stars is somewhat similar to Ursa Major and Ursa Minor. The total mass contained in the cluster is estimated to be about 800 solar masses and is dominated by fainter and redder stars. An estimate of the frequency of binary stars in the Pleiades is about 57%.


title: Pleiades

text: The cluster contains many brown dwarfs, which are objects with less than about 8% of the Sun's mass, not heavy enough for nuclear fusion reactions to start in their cores and become proper stars. They may constitute up to 25% of the total population of the cluster, although they contribute less than 2% of the total mass. Astronomers have made great efforts to find and analyse brown dwarfs in the Pleiades and other young clusters, because they are still relatively bright and observable, while brown dwarfs in older clusters have faded and are much more difficult to study.


title: Pleiades

text: The nine brightest stars of the Pleiades are named for the Seven Sisters of Greek mythology: Sterope, Merope, Electra, Maia, Taygeta, Celaeno, and Alcyone, along with their parents Atlas and Pleione. As daughters of Atlas, the Hyades were sisters of the Pleiades. The following table gives details of the brightest stars in the cluster:


title: Pleiades

text: Ages for star clusters can be estimated by comparing the Hertzsprung–Russell diagram for the cluster with theoretical models of stellar evolution. Using this technique, ages for the Pleiades of between 75 and 150 million years have been estimated. The wide spread in estimated ages is a result of uncertainties in stellar evolution models, which include factors such as convective overshoot, in which a convective zone within a star penetrates an otherwise non-convective zone, resulting in higher apparent ages.


title: Pleiades

text: Another way of estimating the age of the cluster is by looking at the lowest-mass objects. In normal main-sequence stars, lithium is rapidly destroyed in nuclear fusion reactions. Brown dwarfs can retain their lithium, however. Due to lithium's very low ignition temperature of 2.5 × 10 K, the highest-mass brown dwarfs will burn it eventually, and so determining the highest mass of brown dwarfs still containing lithium in the cluster can give an idea of its age. Applying this technique to the Pleiades gives an age of about 115 million years.


title: Pleiades

text: The cluster is slowly moving in the direction of the feet of what is currently the constellation of Orion. Like most open clusters, the Pleiades will not stay gravitationally bound forever. Some component stars will be ejected after close encounters with other stars; others will be stripped by tidal gravitational fields. Calculations suggest that the cluster will take about 250 million years to disperse, with gravitational interactions with giant molecular clouds and the spiral arms of our galaxy also hastening its demise.


title: Pleiades

text: With larger amateur telescopes, the nebulosity around some of the stars can be easily seen; especially when long-exposure photographs are taken. Under ideal observing conditions, some hint of nebulosity around the cluster may even be seen with small telescopes or average binoculars. It is a reflection nebula, caused by dust reflecting the blue light of the hot, young stars.


title: Pleiades

text: It was formerly thought that the dust was left over from the formation of the cluster, but at the age of about 100 million years generally accepted for the cluster, almost all the dust originally present would have been dispersed by radiation pressure. Instead, it seems that the cluster is simply passing through a particularly dusty region of the interstellar medium.


title: Pleiades

text: Studies show that the dust responsible for the nebulosity is not uniformly distributed, but is concentrated mainly in two layers along the line of sight to the cluster. These layers may have been formed by deceleration due to radiation pressure as the dust has moved towards the stars.


title: Pleiades

text: Analyzing deep-infrared images obtained by the Spitzer Space Telescope and Gemini North telescope, astronomers discovered that one of the cluster's stars, HD 23514, which has a mass and luminosity a bit greater than that of the Sun, is surrounded by an extraordinary number of hot dust particles. This could be evidence for planet formation around HD 23514.


title: Great Resignation

text: The Great Resignation, also known as the Big Quit and the Great Reshuffle, is an ongoing economic trend in which employees have voluntarily resigned from their jobs "en masse," beginning in early 2021 in the wake of the COVID-19 pandemic. Among the most cited reasons for resigning include wage stagnation amid rising cost of living, limited opportunities for career advancement, hostile work environments, lack of benefits, inflexible remote-work policies, and long-lasting job dissatisfaction. Most likely to quit have been workers in hospitality, healthcare, and education.


title: Great Resignation

text: Some economists have described the Great Resignation as akin to a general strike. However, workforce participation in some regions has recovered or even exceeded the pre-pandemic rate. This suggests that instead of remaining out of the workforce for extended periods (which can be financially difficult, especially at a time of high inflation), many workers have been simply swapping jobs.


title: Great Resignation

text: The term "Great Resignation" was coined by Anthony Klotz, a professor of management at University College London's School of Management, in May 2021, when he predicted a sustained mass exodus.


title: Great Resignation

text: Between December 2000, when quit rates were first measured by the United States Department of Labor, and February 2021, roughly a year following the beginning of the COVID-19 pandemic, the U.S. resignation rate never surpassed 2.4% of the total workforce per month. High quit rates indicate worker confidence in the ability to get higher paying jobs, which typically coincides with high economic stability and low unemployment rates. Conversely, during periods of high unemployment, resignation rates tend to decrease as hire rates also decrease. For example, during the Great Recession, the U.S. quit rate decreased from 2.0% to 1.3% as the hire rate fell from 3.7% to 2.8%.


title: Great Resignation

text: Resignation rates in the U.S. during the pandemic initially followed this pattern. In March and April 2020, a record 13.0 and 9.3 million workers (8.6% and 7.2%) were laid off, and the quit rate subsequently fell to a seven-year low of 1.6%. Much of the layoffs and resignations were driven by women, who disproportionately work in industries that were affected most by the lockdowns, like service industries and childcare.


title: Great Resignation

text: As the pandemic continued, however, workers began to quit their jobs in large numbers despite initially high unemployment.


title: Great Resignation

text: According to Microsoft's "Work Trend Index", more than 40% of the global workforce were considering quitting their job in 2021. The COVID-19 pandemic allowed workers to rethink their careers, work conditions, and long-term goals. As many workplaces attempted to bring their employees in-person, workers desired the freedom that remote work afforded them during the COVID-19 pandemic, as well as schedule flexibility, which was the primary reason to look for a new job of the majority of those studied by Bankrate in August 2021. Additionally, many workers, particularly in younger cohorts, are seeking to gain a better work–life balance. In the U.S., connections are also being drawn to reported increases in workplace stress and employee burnout. Moreover, millions of people now have long COVID; this disability can alter the ability or desire to work.


title: Great Resignation

text: Restaurants and hotels, industries that require in-person interactions, have been hit the hardest by waves of resignations. COVID-19 stimulus payments and rises in unemployment benefits allowed those who relied on low-wage jobs for survival to stay home, although places where unemployment benefits were rolled back did not see significant job creation as a result. On the other hand, many workers who are dissatisfied with their jobs report that they cannot resign due to economic barriers, many of these workers being people of color. Sekou Siby, president and CEO of the U.S. nonprofit Restaurant Opportunities Center United, commented, "There's more competition across industries, so workers are feeling more empowered than ever before, but that doesn't mean everyone is able to leave their current jobs."


title: Great Resignation

text: According to a study conducted by Adobe, the exodus is being driven by Millennials and Generation Z, who are more likely to be dissatisfied with their work. More than half of Gen Z reported planning to seek a new job within the next year. "Harvard Business Review" found that the cohort between 30 and 45 years old had the greatest increase in resignation rates. Racial minority, low-wage, and frontline workers are also more dissatisfied with their work in the United States, according to the asset management firm Mercer.


title: Great Resignation

text: An IMF working paper by Carlo Pizzinelli and Ippei Shibata focused on causes of the loss in employment within the U.S. and U.K. labor markets in comparison to pre-COVID-19 levels. They identified job mismatch (that is, mismatch between the areas where people search for work and where the most vacancies are) as playing a "modest" role, being less significant than in the wake of the global financial crisis. The effect of the pandemic on women, sparking the so-called "She-cession", was deemed as accounting for some 16% of the total U.S. employment shortfall but little to none of the shortfall in the U.K. Meanwhile, the authors attribute 35% of the shortfall in both the U.K. and U.S. to older workers (aged 55–74) withdrawing from the labor force. On the other side, "Harvard Business Review" reported resignation rates for those aged 60–70 actually fell in 2021 (in comparison to 2020 rates).


title: Great Resignation

text: Some suggest that the tightness of the labor market is not due to workers resigning "en masse" or turning towards self-employment, but instead due to a shortage of migrant workers caused by travel restrictions during the pandemic and a general rise in anti-immigrant sentiment.


title: Great Resignation

text: There is disagreement as to whether or not the Great Resignation will have a lasting effect. Although quit rates remain high, many workers in Western countries have been returning to the labor force in large numbers in late 2021 and early 2022. In general, an aging population and a labor shortage accelerate industrial automation, not just to replace the "missing" workers but also to cut costs.


title: Great Resignation

text: In February 2022, Australian treasurer Josh Frydenberg reported that the labor market had been experiencing a “Great Reshuffle” rather than a “Great Resignation”. He also reported that over one million workers started new jobs in the three months prior to November 2021, an increase of almost 10% prior to the pre-pandemic average. In the three months prior to February 2022, 300,000 workers reported resigning for better job opportunities, a record number. A main incentive may have been higher pay, as the typical Australian worker who switched jobs received a pay bump of 8% to 10%.


title: Great Resignation

text: A similar social protest movement is occurring in China, referred to as "tang ping" (). It started roughly during the same time as the Great Resignation, in April 2021. It is a rejection of societal pressures to overwork, such as in the 996 working hour system. Those who participate in "tang ping" instead choose to "lie down flat and get over the beatings" via a low-desire, more indifferent attitude towards life. Business magazine "ABC Money" claimed the lifestyle resonated with youth disillusioned by the government-endorsed "Chinese Dream", which encourages a life of hard work and sacrifice without life satisfaction to show for it. The Chinese Communist Party (CCP) has worked to reject the idea through state-owned media and internet censorship, though some party voices offer that the movement provides an opportunity to reflect on how best to cultivate diligence in young generations.


title: Great Resignation

text: A survey by HR company SD Worx of 5,000 people in Belgium, France, the U.K., Germany, and the Netherlands, found that employees in Germany had the most COVID-19-related resignations, with 6.0% of the workers leaving their jobs. This was followed by the United Kingdom with 4.7%, the Netherlands with 2.9%, and France with 2.3%. Belgium had the least number of resignations with 1.9%.


title: Great Resignation

text: Some preliminary data show an increase in the number of quits in Italy, starting in the second quarter of 2021. The registered increase was not only in absolute terms, but also in terms of quit rate (computed as quits over employed population) and of quit share (computed as quits over total contract terminations).


title: Great Resignation

text: In the United Kingdom between July and September 2021, over 400,000 workers left their jobs, up from 270,000 two years prior. There were a record high 1.3 million job vacancies in December 2021, or 4.4 vacancies for every 100 jobs. The U.K. workforce got smaller in 2020, the first time in over twenty years, largely due to older people retiring according to Tony Wilson, director of the Institute for Employment Studies. Between October and December 2021, the rate of U.K. workers aged 16 to 64 moving job-to-job was at an all-time high of 3.2%. Workers in their 30s (Millennials) are the most likely to quit their jobs, followed by those in their 20s (Generation Z). However, by August 2022, many British workers have returned to their previous positions after quitting and some elderly Britons have opted out of retirement in order to pay their bills in the wake of high inflation.


title: Great Resignation

text: Some industries, like nursing, have been hit especially hard by burnout. In October 2021, before the Omicron variant caused a new wave of cases, the Royal College of Nursing conducted a survey of over 9,000 British nursing staff in which 57% of respondents were either thinking about leaving or actively planning to leave their jobs. The primary factors reported were feeling undervalued, exhausted, and not being able to give adequate care.


title: Great Resignation

text: India has witnessed large scale resignations across many sectors of the economy. The information technology sector in particular witnessed massive attrition, with over a million resignations in 2021.


title: Great Resignation

text: In April 2021, a record 4.0 million Americans quit their jobs. In June 2021, approximately 3.9 million American workers quit their jobs. Resignations are consistently the most prevalent in the South, where 2.9% of the workforce voluntarily left their jobs in June, followed by the Midwest (2.8%) and the West (2.6%). The Northeast is the most stable region, with 2.0% of workers quitting in June.


title: Great Resignation

text: According to a PricewaterhouseCoopers survey conducted in early August 2021, 65% of employees said they were looking for a new job and 88% of executives said their company was experiencing higher turnover than normal. A Deloitte study published in "Fortune" magazine in October 2021 found that among Fortune 1000 companies, 73% of CEOs anticipated the work shortage would disrupt their businesses over the next 12 months, 57% believed attracting talent is among their company's biggest challenges, and 35% already expanded benefits to bolster employee retention.


title: Great Resignation

text: Workers in the leisure and hospital industries had relatively high quit rates. In October 2021, the U.S. Bureau of Labor Statistics reported that food service workers' quit rates rose to 6.8%, which is well above the industry average of 4.1% over the last 20 years and still higher than the industry's quit peaks of 5% in 2006 and 2019. The retail industry had the second highest quit rates at 4.7%. From the start of the pandemic to November 2021, approximately one in five healthcare workers quit their jobs.


title: Great Resignation

text: Amidst the Great Resignation, October 2021 saw a strike wave known as Striketober, with over 100,000 American workers participating in or preparing for strike action. While discussing Striketober, some economists described the Great Resignation as workers participating in a general strike against poor working conditions and low wages.


title: Great Resignation

text: By August 2022, however, the U.S. workforce has surpassed its size from before the COVID-19 pandemic. Many American workers took advantage of the labor shortage to trade their current jobs for those with higher salaries, more benefits, and better schedules. Some have started or joined labor unions. A number of prospective employers are also offering paid training in order to attract recruits. Still, the recovery of employment has been uneven, as low-wage sectors—especially leisure, hospitality, retail, manufacturing, and education—lose jobs to those that offer higher income. Although retirees (who need to supplement their incomes) have returned at a rate not seen since 2019, it remains unclear whether workers aged 55 and over, people more likely to quit, will ever return. Unsatisfied with their current positions, a record number of Americans quit to start their own businesses, only to face an acute labor shortage they helped to create. In fact, small businesses are the most likely to struggle to find qualified recruits. Some 16 million Americans suffer from long COVID, and of these, 2 to 4 million are kept out of work because if it. The shortage of workers has exacerbated the disruption of the domestic supply chain of the United States.


title: Great Resignation

text: In response to the problem, a number of firms have relocated to states with lower costs of doing business (and possibly with subsidies), a large pool of skilled workers, quality education, high standards of living, and good infrastructure. Wage growth has jumped; in December 2021, wage growth reached 4.5%, the highest since June 2001. Some employers in the fast food industry, like McDonalds, are providing more benefits, like college scholarships and healthcare benefits, to bring back workers. Public sector jobs have had higher worker retention as compared to private sector jobs, largely due to stronger benefits like paid family leave.


title: Great Resignation

text: Productivity has dropped during the Great Resignation because even when employees stay, they are not as productive as they were in the past.


title: Great Resignation

text: Fans and media outlets drew connections between the Great Resignation and Beyoncé's 2022 song "Break My Soul", specifically in the verse "Now I just fell in love / And I just quit my job".


title: Sterling K. Brown

text: Sterling Kelby Brown (born April 5, 1976) is an American actor. He has portrayed Christopher Darden in the FX limited series "" (2016) and stars as Randall Pearson in the NBC drama series "This Is Us" (from 2016-2022). Both roles have earned him Primetime Emmy Awards and the latter also won him a Golden Globe Award. He has also had supporting roles in the films "Black Panther" (2018) and "Waves" (2019), and recently appeared on the Amazon Prime original series "The Marvelous Mrs. Maisel". He has also voiced characters in "The Angry Birds Movie 2" and "Frozen II", both released in 2019. He was included in "Time" magazine's list of 100 Most Influential People of 2018"."


title: Sterling K. Brown

text: Brown was born in St. Louis, Missouri to Sterling Brown and Aralean (Banks) Brown. He has two sisters and two brothers. His father died when he was 10 years old.


title: Sterling K. Brown

text: As a child, he went by the name Kelby; when he turned 16, he adopted the name Sterling, explaining in 2016:


title: Sterling K. Brown

text: Brown grew up in Olivette, Missouri, an inner-ring suburb of St. Louis. He attended the private Mary Institute and St. Louis Country Day School.


title: Sterling K. Brown

text: He graduated from Stanford University in 1998 with an acting degree. He had initially planned to major in economics with a focus on business, but fell in love with acting as a freshman. He did post-graduate study at New York University's Tisch School of the Arts, where he graduated with a Master of Fine Arts degree.


title: Sterling K. Brown

text: After college, Brown performed a series of roles in regional theater. He has also appeared on numerous television shows including "ER", "NYPD Blue", "JAG", "Boston Legal", "Alias", "Without a Trace", "Supernatural", and "Third Watch". He was a regular in the comedy "Starved", and has also appeared in movies, including "Stay" with Ewan McGregor, "Brown Sugar" with Taye Diggs, and "Trust the Man" with David Duchovny and Julianne Moore.


title: Sterling K. Brown

text: He had a recurring role on the television series "Supernatural", as vampire hunter Gordon Walker. He played Dr. Roland Burton on "Army Wives" and Detective Cal Beecher on "Person of Interest", and appeared on "Medium". In 2008, he played David Mosley on the "Patience" episode of "Eli Stone". In 2016, he starred in the FX miniseries "" as Christopher Darden, for which he won the Primetime Emmy Award for Outstanding Supporting Actor in a Limited Series or Movie at the 68th Primetime Emmy Awards.


title: Sterling K. Brown

text: In the theater, Brown was cast in the 2002 production of Bertolt Brecht's "The Resistible Rise of Arturo Ui" starring Al Pacino, Paul Giamatti, Steve Buscemi, John Goodman and Jacqueline McKenzie. In 2014, he starred as Hero in Suzan-Lori Parks' "Odyssey"-inspired play "Father Comes Home From the Wars" at New York's Public Theater. He also starred in the 2014 movie "The Suspect" with Mekhi Phifer.


title: Sterling K. Brown

text: Since 2016, Brown has had a starring role in the television series "This Is Us". In 2018, it made him the first African-American actor to win a Golden Globe in the Best Actor in a Television Drama category, and the first to win a Screen Actors Guild Award in the Outstanding Performance by a Male Actor in a Drama Series category. He also won, with the rest of the cast, the Screen Actors Guild Award for Outstanding Performance by an Ensemble in a Drama Series. That year he also appeared as N'Jobu in the Marvel Cinematic Universe film "Black Panther".


title: Sterling K. Brown

text: He appeared as a guest star in the NYPD sitcom "Brooklyn Nine-Nine", and earned critical acclaim for his role as Philip Davidson, a dentist under investigation for murdering his business partner. He was nominated for an Primetime Emmy Award for Outstanding Guest Actor in a Comedy Series in 2018 for this performance.


title: Sterling K. Brown

text: In August 2019, he was announced at D23 Expo and on Twitter as the voice of Lieutenant Destin Mattias in "Frozen II".


title: Sterling K. Brown

text: Brown met actress Ryan Michelle Bathe as a college freshman at Stanford. They eloped in March 2006, but held a large ceremony in June 2007. They have two sons, Andrew and Amaré.


title: Birthday problem

text: In probability theory, the birthday problem asks for the probability that, in a set of randomly chosen people, at least two will share a birthday. The birthday paradox is that, counterintuitively, the probability of a shared birthday exceeds 50% in a group of only 23 people.


title: Birthday problem

text: The birthday paradox is a veridical paradox: it appears wrong, but is in fact true. While it may seem surprising that only 23 individuals are required to reach a 50% probability of a shared birthday, this result is made more intuitive by considering that the comparisons of birthdays will be made between every possible pair of individuals. With 23 individuals, there are (23 × 22) / 2 = 253 pairs to consider, much more than half the number of days in a year.


title: Birthday problem

text: Real-world applications for the birthday problem include a cryptographic attack called the birthday attack, which uses this probabilistic model to reduce the complexity of finding a collision for a hash function, as well as calculating the approximate risk of a hash collision existing within the hashes of a given size of population.


title: Birthday problem

text: The problem is generally attributed to Harold Davenport in about 1927, though he did not publish it at the time. Davenport did not claim to be its discoverer "because he could not believe that it had not been stated earlier". The first publication of a version of the birthday problem was by Richard von Mises in 1939.


title: Birthday problem

text: From a permutations perspective, let the event be the probability of finding a group of 23 people without any repeated birthdays. Where the event is the probability of finding a group of 23 people with at least two people sharing same birthday, . is the ratio of the total number of birthdays, formula_1, without repetitions and order matters (e.g. for a group of 2 people, mm/dd birthday format, one possible outcome is formula_2 divided by the total number of birthdays with repetition and order matters, formula_3, as it is the total space of outcomes from the experiment (e.g. 2 people, one possible outcome is formula_4. Therefore formula_1 and formula_3 are permutations.


title: Birthday problem

text: Another way the birthday problem can be solved is by asking for an approximate probability that in a group of people at least two have the same birthday. For simplicity, leap years, twins, selection bias, and seasonal and weekly variations in birth rates are generally disregarded, and instead it is assumed that there are 365 possible birthdays, and that each person's birthday is equally likely to be any of these days, independent of the other people in the group. For independent birthdays, the uniform distribution on birthdays is the distribution that minimizes the probability of two people with the same birthday; any unevenness increases this probability. The problem of a non-uniform number of births occurring during each day of the year was first addressed by Murray Klamkin in 1967. As it happens, the real-world distribution yields a critical size of 23 to reach 50%.


title: Birthday problem

text: The goal is to compute , the probability that at least two people in the room have the same birthday. However, it is simpler to calculate , the probability that no two people in the room have the same birthday. Then, because and are the only two possibilities and are also mutually exclusive,


title: Birthday problem

text: Here is the calculation of for 23 people. Let the 23 people be numbered 1 to 23. The event that all 23 people have different birthdays is the same as the event that person 2 does not have the same birthday as person 1, and that person 3 does not have the same birthday as either person 1 or person 2, and so on, and finally that person 23 does not have the same birthday as any of persons 1 through 22. Let these events be called Event 2, Event 3, and so on. Event 1 is the event of person 1 having a birthday, which occurs with probability 1. This conjunction of events may be computed using conditional probability: the probability of Event 2 is 364/365, as person 2 may have any birthday other than the birthday of person 1. Similarly, the probability of Event 3 given that Event 2 occurred is 363/365, as person 3 may have any of the birthdays not already taken by persons 1 and 2. This continues until finally the probability of Event 23 given that all preceding events occurred is 343/365. Finally, the principle of conditional probability implies that is equal to the product of these individual probabilities:


title: Birthday problem

text: This process can be generalized to a group of people, where is the probability of at least two of the people sharing a birthday. It is easier to first calculate the probability that all birthdays are "different". According to the pigeonhole principle, is zero when . When :


title: Birthday problem

text: The equation expresses the fact that the first person has no one to share a birthday, the second person cannot have the same birthday as the first (), the third cannot have the same birthday as either of the first two (), and in general the th birthday cannot be the same as any of the preceding birthdays.


title: Birthday problem

text: The event of at least two of the persons having the same birthday is complementary to all birthdays being different. Therefore, its probability is


title: Birthday problem

text: The following table shows the probability for some other values of (for this table, the existence of leap years is ignored, and each birthday is assumed to be equally likely):


title: Birthday problem

text: According to the approximation, the same approach can be applied to any number of "people" and "days". If rather than 365 days there are , if there are persons, and if , then using the same approach as above we achieve the result that if is the probability that at least two out of people share the same birthday from a set of available days, then:


title: Birthday problem

text: The probability of any two people not having the same birthday is . In a room containing "n" people, there are pairs of people, i.e. events. The probability of no two people sharing the same birthday can be approximated by assuming that these events are independent and hence by multiplying their probability together. In short can be multiplied by itself times, which gives us


title: Birthday problem

text: Since this is the probability of no one having the same birthday, then the probability of someone sharing a birthday is


title: Birthday problem

text: The result is over 50% as previous descriptions. This approximation is the same as the one above based on the Taylor expansion that uses formula_23.


title: Birthday problem

text: which works well for probabilities less than or equal to . In these equations, is the number of days in a year.


title: Birthday problem

text: This can also be approximated using the following formula for the "number" of people necessary to have at least a chance of matching:


title: Birthday problem

text: This is a result of the good approximation that an event with probability will have a chance of occurring at least once if it is repeated times.


title: Birthday problem

text: The lighter fields in this table show the number of hashes needed to achieve the given probability of collision (column) given a hash space of a certain size in bits (row). Using the birthday analogy: the "hash space size" resembles the "available days", the "probability of collision" resembles the "probability of shared birthday", and the "required number of hashed elements" resembles the "required number of people in a group". One could also use this chart to determine the minimum hash size required (given upper bounds on the hashes and probability of error), or the probability of collision (for fixed number of hashes and probability of error).


title: Birthday problem

text: For comparison, to is the uncorrectable bit error rate of a typical hard disk. In theory, 128-bit hash functions, such as MD5, should stay within that range until about documents, even if its possible outputs are many more.