Birthday Problem

In: Science

Submitted By fogelly
Words 2471
Pages 10
Birthday Problem

Subject: Mathematics
Name:Dmitry Kozyrev
Passport Number:N07797989
Date: 16.06.2015

Abstract _______________________________________________ 3
Keywords______________________________________________ 3
1. Basis of Probability ___________________________________ 4
1.1 Conditional Probability _____________________________ 4
1.2 Independence _____________________________________ 4
2. Birthday Problem______________________________________5
2.1 What is Birthday Problem? _________________________ 5
2.2 Understanding the probability_________________________ 6
2.3 Calculating the probability of birthday problem___________ 7
2.4 Abstract proof______________________________________10
3 Examples______________________________________________11
4 References_____________________________________________13


The subject of probability can be traced back to the 17th century when it
arose out of the study of gambling games. As we see, the range of applications
extends beyond games into business decisions, insurance, law, medical tests, and
the social sciences. The stock market, “the largest casino in the world,” cannot do
without it. The telephone network, call centers, and airline companies with their
randomly fluctuating loads could not have been economically designed without
probability theory. To quote Pierre-Simon, marquis de Laplace from several
hundred years ago:
«It is remarkable that this science, which originated in the consideration of
games of chance, should become the most important object of human
knowledge...The most important questions of life are, for the most part, really only
problems of probability.»

Keywords: experiment; outcomes; empty set; union; intersection; probability;
birthday problem.


1. Basis of probability
1.1 Conditional Probability
Conditional Probability is…...

Similar Documents

A Birthday Remembered

...A Birthday Remembered The story takes place at Ellen’s a year after Jackie passed away. The situation is that when Jackie died, Tobie’s father got the custody of her, but because he has a problem in dealing with the fact that Tobie’s mother was a lesbian, he won’t let Tobie see Ellen, who’s the woman Jackie had been in a relationship with. After a year living at her fathers, Tobie calls Ellen at her birthday to congratulate, and invites herself and a friend for a cup of cocoa. Following they talk about Jackie and love. In the story, a third person narrator appears, and it is seen from Ellen’s point of view. Ellen is a lesbian, and she had had a relationship with Jackie, who passed away about a year earlier. She is born on Valentine’s Day and she had become editor of the food section and had written a cookbook. At the time, when Jackie passed away, and Tobie were to live with her father, she was very sad. She did not get a chance to say goodbye to Jackie during the last rituals, because of the father, and that has made a deep wound inside of her. She does not feel like Tobie belongs to her – them - anymore, and she miss both of them very much. Now she has moved on a bit, and can look back and recall without too much pain, but there is many things about Tobie, that reminds her of Jackie and the past. Tobie is a girl with blond hair, which is cut short, and physically she looks like her father with the sharp, angular face. She looks like her mother in the warmth and...

Words: 446 - Pages: 2

21st Birthday

...My 21st birthday was a wonderful day full of love, attention and excitement. Birthday breakfasts are family traditions that have been around all my life. They are always only family. Omelets are my favorite breakfast food. My mother makes the lightest, fluffiest eggs I have ever had. She always makes your personal breakfast a great big deal. Extravagance is abundant at these events. Purple is my favorite color so all the decorations are purple. Dishes are purple with purple flowers as garnishes. Breakfast meats, such as bacon, sausage, ham and Canadian bacon, are on large platters all over the table. Of course the smoky, sweet meaty essences of smells are hanging in the air. Mixed into this meaty smell is a sweet cinnamon fragrance of the sweet rolls. At our family gatherings alcohol is ever present. Orange juice and champagne, otherwise known as mimosas, are prevalent. Mom always brings out the beautiful crystal champagne flutes to use. I always feel so special drinking out of these glasses. My two sisters are there. They came in from out of town for the occasion. I don’t get to see them often anymore so it was a special treat that they were there. After the meal we do the gifts and presents opening. We always try to get the person something they can use but wouldn’t get for themselves. My mom gets me a gift card to a spa for a day. It is awesome and includes a facial, massage, pedicure and manicure. My sisters have gotten me a gift card for the mall...

Words: 1218 - Pages: 5

Birthday Headline

...Birthday Headlines Pauline F. Bacay De La Salle – College of St. Benilde Introduction As final requirement for Technical Writing class, the students were to research on events that were headlines on the day they were born, 20 years, 50 years and 100 years ago. They were to present this in front of a panel in specific schedules their professor has given. This also served as their final exam for the second term of school year 2012 – 2011. They were to follow the APA style in doing the paper including a bibliography of the same format. The research is very useful in their course, which is Bachelor of Arts Major in Consular and Diplomatic Affairs, because it concerns historical headlines around the world and their course mostly consists of majors in history. They may also improve their research making skills and strategies in the study because it requires them to consult more sources for a precise and reliable study. Based on this study, many historical events did happen on the same month of the researcher’s birthday. Some created great effect on today and others are just headlines that shook the world. It’s a way of discovering new things about history and also getting to remember it because it took a mindful of research to get all the information. It also serves as an achievement for a student to create a research paper and applying all that they have learned in their subject. BODY January 31, 1996 50 Dead in Sri Lanka suicide bombing BBC News UK (1996...

Words: 2918 - Pages: 12

Happy Birthday

...Write a summary of Happy Birthday, 1951 in about 150 words."Happy Birthday, 1951" is a short story by Kurt Vonnegut. The main characters in the shortstory are a man and a boy. We are told that a refugee woman left her baby by the old manand then she never came back - That's how the man got the boy. The man doesn't knowsomething about the boy and that's why they are choosing a day to celebrate the boy'sbirthday. As a birthday present the man wants to take the boy to a place without war. Ontheir way to the place, they talk about the color of different uniforms when they see foursoldiers. When they arrive, they eat their lunch, close their eyes and sleep for a while. In theend of the story, we are told that the man wakes up and discovers that the boy is gone.Different thoughts go through the man's head and those are interrupted by the boy when hesays "Gotcha!"Characterize the boy and the old man.The two main characters in the short story are the boy and the old man. The old man in the shortstory feels that he hasn't been a good father, which is expressed different places in the story, suchas:"I haven't been a good father, letting you go without birthdays this long… You're entitled toone every year, you know, and I've let six years go by without a birthday. And presents, too. Youare supposed to get presents." (7. 30 - 32)The old man does not like the war and he does not like the soldiers. Every time the boy talks aboutsomething which has to do with the war, the old man gets a bit......

Words: 291 - Pages: 2

Birthday Letters

...Ted Hughes • Hughes’ destruction of Plath’s diaries ( influenced by Plath’s own morbid curiosity for her dead parent. Fulbright Scholars • 1st poem in Birthday Letters. • Personal, a ‘letter’ to plath. Personal pronoun unlike Hughes other poems – hughe’s rarely used 1st person. • Personal pronouns: “you,” “I,” “your,” “my” • Harsh “i” alliteration, “it,” – trying to pinpoint exact memories. • Questions: Could be a critisism of the public obsession with his life ie. He is asking US, or could be questioning his own hazy memory. • Conjectures – ability to tell the truth is limited by our own knowledge. • Alliteration “m” sounds: mumbling,confused, remembering. • Peach || biblical apple. ( symbolic of loss of innocence. • 1 stanza – stream of consciousness • The haziness of the memory gives the impression that hughes is being truthful, although gives the impression of incomplete or disjointed truths. Also allows hughes to utilise creative license. • Juxtaposition of initial haziness with final, clear, peach image. Precise image, if hughe’s can remember the peach he is being truthful about not recalling the photograph. • Appearance vs reality, as when hughe’s described plath’s appearance – “your veronica lake bang – not what it hid.” The picture presents one image but hughes “knows” the “truth.” • Photograph – photographs only show their subjects from the outside: appearance vs...

Words: 1120 - Pages: 5

Happy Birthday’’

...1. Write a summary of ‘’Happy birthday’’ 1951 in about 150 words. ‘’Happy birthday’’ it’s a text about an old man and a young boy, who live in a ruin, in a former warzone. The city is still full of soldiers and tanks. The man got the baby boy about six years ago from a refugee woman. The story begins with the man wanting the boy to choose a birthday. The boy chooses the following day. The old man puts together a cart for the boy, but he also wants the boy to experience a world without war. Next day the boy gets the cart and pretends its a tank, because he is fascinated by war, but the old man want the boy to pretend that it is a truck. After that the man takes the boy into the forest for a day to experience the world as it should be. They pass a tank on their way, and the boy is all ecstatic about it, but the man wants the boy to come along into the forest. The man falls asleep, and when he wakes up late in the afternoon then he found out that the boy was missing. The old man is calling the boy and in the end the boy comes out from the tank 2. Characterize the boy and the old man. The old mand: The old man is kind, friendly and helpful, because he took the little baby without any expectation to get something back, like money or help from others. He spawned the little boy with his money, no help from others and his pure love like a father. The old man does not like war and the soldiers. He wants to take the boy away from war, because he thinks that the boy is too...

Words: 544 - Pages: 3

Birthday Cake

... experiencing nostalgia, reflecting on past birthdays when he was happier and younger. This is the only positive connection Goodman makes with the birthday cake. Unfortunately, this tone is reserved only for the second stanza because frustration takes over in the first and third stanzas. Furthermore, Goodman shows a nostalgic, emotional tone in the second stanza as opposed to the frustrated tone in the first. The latter consists of short, abrupt verses consisting of cacophony. The former consists of euphonic verses with the lovely, as Goodman describes it, image of a lit birthday cake ready to be presented with a song to the speaker, despite the underlining dark message within the dark room. The cacophony and constant reminder of too many, too many, too many candles in the first stanza reveal the frustration of the speaker. It almost seems as if the speaker is stammering with frustration, like an elderly person would stammer. The second stanza uses longer verses with euphonic words such as “fluttering,” and “most loveliest sight.” These words possess harmony and easily roll off the tongue when read aloud. The first stanza’s “too many to blow out / too many to count too many” are significantly harsher on the ears. Next, the sestet poem turns to a frustrated tone in the third, and final, stanza with a metaphorical solution to Goodman’s age problem. Goodman uses anaphora once more with phrases from the first stanza: “isn’t it time” repeated twice and “too......

Words: 1012 - Pages: 5

Happy Birthday

...happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday happy birthday...

Words: 572 - Pages: 3

My Birthday Slip

...My Birthday Slip This was my seventeenth birthday and I couldn’t wait for it, although I was going to be in school for it. I was looking forward to all of the birthday wishes, hugs, and gifts. It always happened that way every year, so I shouldn’t have been that much excited; I guess it was because I was turning a year older. The morning of my birthday, I had gotten up and picked out my birthday outfit. I picked skinny leg khaki pants, a white collared aeropostle, shirt, and brown baby doll shoes. To let it be known that it was my birthday, I added a pink birthday stash that read “Birthday Queen” and a pink and silver crown. I made my hair so pretty, with little tight curls and even did my makeup a little. I like to go all out, so of course I had a brand new brown and cheetah print backpack that was similar to my brown baby doll shoes. I was ready to go. By time I made it to my bus stop down the street from my house, I had already received 3 different sets of balloons and a couple of gifts. The whole ride to school I got phone calls and text messages, so I was beyond happy! All of my balloons were in the aisle and in everyone’s way so it was also a difficult ride to school. Still, I didn’t care because it was my birthday. I couldn’t wait to get to school for two reasons: first, to get off the bus with all of my balloons because I started to get a bit annoyed with them and second, to get more birthday wishes, gifts, and attention. Finally, I had made it to school. I...

Words: 556 - Pages: 3

A Birthday Remembered

 People can and will find love many times in their lives. 
 Not only the teenage-love, where they go all blushing and feel the butterflies swirling around in their stomachs. Also the kind of love, which causes a lot of pain. 
 That kind of pain which will grow inside of them and be like an attached leech, which will never let go.
 It is something that hurts a lot, just like the pain love can give us. And force us to feel and show. 
 People are forced to show their emotions. Confront them. Feel them. 
 This will somehow cause the trouble inside that human being the love is haunting, and in the worst scenarios even cause a pain that will just keep growing. 
 A pain, which will just get bigger the more the person notices it. 
 This is something we can see in the main character of the short story ‘’A Birthday Remembered.’’ 
 Ellen can be seen as a woman who has accepted the way she loves other people. She is a woman, who likes other women and that has caused her some problems in her life. 
 People live on the prejudices for lesbians and gay people in general, which can be the kind of main reason for some of her problems. 
 Ellen has found her one and true love, called Jackie. 
 Jackie left her husband so she could be with Ellen, which caused them some more problems now with the ex husband, Roger, nipping on their heels. 
 Which we can see in this quotation ‘’The three of them had survived through the tumultuous stress of trying to make it, ever...

Words: 637 - Pages: 3


...Objective Obtain and maintain a position in which my strong organizational skills, ability to work well with others, outstanding customer service skills, broad educational background, and exceptional leadership skills may be exercised and contribute to the company’s growth. Profile Highly motivated, personable customer service/clerical professional with a good track and a talent for quick learning as well as assuming leadership. Skills Summary Leadership skillsStrong organizational skillsTime management | Computer SkillsCustomer Service Floor OperationsBilingual (Spanish) | * | Professional Experience * Oakley January ’15 – Present * STOCK MANAGER/ASSISTANT MANAGER * Work with General Manager to improve operations, sales, and profitability. * Maximize profits by controlling labor costs and expenses. * Continuously reviewing and managing team performance. * Support Store Manager in identifying opportunities for commercial advantages and delegating work to staff to ensure manageable day-to-day workload and exceptional in store experience. * Checking and signing invoices and credit notes on a weekly basis. * LIDS November ‘13 – January ‘15 Store Manager Completes store operational requirements by scheduling and assigning employees; following up on work results. Maintains store staff by recruiting, selecting, orientating, and training new employees. Achieves financial objectives by preparing an annual budget...

Words: 364 - Pages: 2

Happy Birthday

...Happy birthday to the biggest treasure in my life! I’m so proud and happy that I have such a beloved, adorable, wonderful, amazing and the most extraordinary guy in the world! You’re always there for me, notwithstanding the distance. You’re the most amazing person I’ve ever met in my life, smart, kind, gentle, simply irreplaceable, just the man of my dreams who came in the nick of time to put a spell on me as well as Prince Charming engages in a quest to liberate a damsel in distress from evil. This special occasion, this day is really unusual not only for you but also for me. Your showing the tender sentiment, growing, lofty and strong, makes me love you more dearly, deeply, passionately and endlessly. All your golden deeds are really invaluable. You are a miracle worker who does marvels. I can’t express the glow of happiness I feel but what I want to wish you is to stride purposefully on the path called “Life” and I promise to follow you and march in the step with you no matter what. The journey of my life is so amazing when you are next to me. I wish you a lifetime of happiness, good health. You’re a guy of many accomplishments, just try to make further progress like an oceangoing ship which is reaching the intended destination. You’re a heaven-born programmer, physicist, trying to help me with everything. If only there were more people of your kind, then the world would be a sheer paradise. But nevertheless I have you, like the sun which shines only to me, the best......

Words: 302 - Pages: 2

Birthday Cake History

...Birthday Cake History In our society today birthdays are one of the celebrations that we hold in a year. Every year a person gets older and we celebrate that factor. The day that the person was born on is the day that we celebrate their birthday. We have party favors, noise makes, food, punch, balloons, presents and cake. So when the person’s birthday is happening everyone is always excited for the cake, but why is the cake so important to us, how did the tradition of the birthday cake get started and what is the deal with the birthday candles? It is said that back in the time of the Greeks they would make these sweet beads and top them off with honey and bring them to the temple and offer them to the gods. In an article that I found this is what they understand “Cakes were initially called by the name ‘plakous’ by the Greeks, which is a similar word to ‘flat.’ These were a combination of nuts and honey. Besides, they also made a cake called "satura," which was a flat heavy cake. It is also believed that the Greeks offered round shaped cake to the Goddess of Moon called Artemis[1].” In another article it talks about the Romans ad how they brought cakes into play, “During Roman times, 'cakes' were served at special birthdays. However, these cakes did not resemble contemporary cakes in the least. They were simply flat rounds made with flour containing nuts, leavened with yeast, and sweetened with honey. In early Europe, the words for cake and bread were...

Words: 1050 - Pages: 5

Happy Birthday to You

..."Happy Birthday to You", also known more simply as "Happy Birthday", is a song that is traditionally sung to celebrate the anniversary of a person's birth. According to the 1998 Guinness Book of World Records, "Happy Birthday to You" is the most recognized song in the English language, followed by "For He's a Jolly Good Fellow". The song's base lyrics have been translated into at least 18 languages.[1], p. 17 The melody of "Happy Birthday to You" comes from the song "Good Morning to All", which was written and composed by American siblings Patty Hill and Mildred J. Hill in 1893.[2][3] Patty was a kindergarten principal in Louisville, Kentucky, developing various teaching methods at what is now the Little Loomhouse;[4] Mildred was a pianist and composer.[1], p. 7 The sisters created "Good Morning to All" as a song that would be easy to be sung by young children.[1], p. 14 The combination of melody and lyrics in "Happy Birthday to You" first appeared in print in 1912, and probably existed even earlier.[1], pp. 31–32 None of these early appearances included credits or copyright notices. The Summy Company registered for copyright in 1935, crediting authors Preston Ware Orem and Mrs. R.R. Forman.[citation needed] In 1990, Warner Chappell purchased the company owning the copyright for $15 million, with the value of "Happy Birthday" estimated at $5 million.[5] Based on the 1935 copyright registration, Warner claims that the United States copyright will not expire until 2030...

Words: 382 - Pages: 2

Happy Birthday

... for small businesses than it will for larger enterprises. 25. Each of these arguments has, at its source, the problem of corruption. These arguments predominantly arise in business environments where facilitation payments are common and will become increasingly difficult arguments to sustain. The Australian Government encourages Australian businesses to work actively to reduce corruption by resisting demands for facilitation payments and reporting them to authorities and works closely with foreign governments to improve governance and reduce corruption. Potential legislative implications 26. The Government is considering whether to remove the defence for facilitation payments by repealing section 70.4 of the Criminal Code, and is seeking submissions from interested parties on this issue. 27. If the facilitation payment defence is removed from the Criminal Code, the Income Tax Assessment Act 1997 (Cth) (the ITAA) would also need to be amended to remove tax deductions for facilitation payments. This could be achieved by repealing sections 26-52(4) and 26-52(5) of the ITAA 1997. Comments are also sought on this issue. When a benefit is legitimately due 28. When considering whether a benefit offered or provided to a person is legitimately due, paragraph 70.2(2)(b) states a court should disregard the value of the benefit. This is to ensure that bribery is an offence irrespective of the value of the benefit offered or provided, as......

Words: 5041 - Pages: 21