# 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
Mark:

Contents
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

2

Abstract
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.

3

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 quickness...

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 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) reported...

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

#### The Birthday Party

...resources for our production of The Birthday Party by Harold Pinter. These resources are divided into 2 parts. Part 1 is intended to give an insight into Pinter, his style of writing and the play itself by looking at previous productions of The Birthday Party. Generally aimed at Higher Drama students, this pack complements units 1, 2 and 3 of the syllabus. However, these resources are also useful for the Knowledge and Understanding elements of Standard Grade as well as the Special Study for Advanced Higher Drama. I have collated information from various sources in order to provide material for use before your students see TAG’s production. In September 2003, Part 2 will be available to download from TAG’s website. This section of the resources will focus on the process of creating our production of The Birthday Party. It will include contributions from the Director, the Designer, the Performers and other members of the creative team. I hope that Part 2 will open up TAG’s working methods and provide a real insight into how a theatre company works. Please feel free to reproduce any section of the on-line resources for your use in the classroom. We are always aiming to provide the most effective and beneficial resources for teachers and so please do offer any comments you have regarding either part of this pack. We welcome and appreciate all feedback. I very much hope that you and your pupils find the resources useful and enjoy TAG’s production of The Birthday Party. Emily......

Words: 7552 - Pages: 31

#### 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

#### Happy Birthday

...Happy Birthday, 1951 by Kurt Vonnegut. Summary: This is a short story about the repercussions of war. The story starts in medias res with an old man choosing the day for a young boy's birthday. There has been a war that ended seven years ago and the pair has been hiding ever since. They were discovered by soldiers and are now filling out their documents. The old man has made a small cart for the boy for his birthday and also takes him for a trip to the countryside. In the calm countryside they discover a rusty tank, which preoccupies the boy a lot. They have lunch and afterwards a small nap. When the old man awakens the boy is gone. He immediately heads for the tank to look for him, fearing he may have drowned in the nearby stream. The boy is nowhere to be seen, but suddenly raises his head from the turret and says “Gotcha!”. (151 words) Characterization of the boy and old man: The boy has never known anything but the war-torn everyday. He has always had to search for food and shelter and has never had a birthday, just as he has missed out on a lot of activities that normal seven-year-olds participate in. He is very preoccupied with the war and the marching soldiers. His birthday gift, which the old man imagines is a grocery cart, the boy pretends to be a tank. Since the boy has never experienced anything but the post-war years he glorifies the soldiers. He has never experienced the gory and bloody truth of war, or known what soldiers do in war. Hence he......

Words: 774 - Pages: 4

#### A Birthday Remembered

...11-11-2013 A Birthday Remembered Homosexuality is not always looked at nicely most men thinks that it is wrong and they don´t know why they just think it is gross and when a woman leaves a man for another woman he thinks that it is wrong and that they are insane. But we are all just people and we have the right to choose who we love. These themes can be seen in the text “A Birthday Remembered” by Ann Allen Shockley from 1980. The story takes place in Ellen´s home over an afternoon until late afternoon, she is in the upper middle class because of her work as an editor for the food section. There isn´t much description of the house, we only read about the living room the kitchen which is spacious and the cellphone. It is where the story begins and ends. Ellen is a strong confident woman who has raised a child “Tobie” with Jackie. She is sad for the loss of Jackie, she is the only mother figurer for Tobie who have lost her mother “Jackie” (“ may I come over?” why shouldn´t she? Then again, why should she really want to? Tobie no longer belonged to her- them. When Jackie died a year ago, Tobie had to go to her father.)ll 7-9. She is a caring woman who is full of love for Tobie. Tobie is like a daughter to her, she is happy that Tobie remembers her birthday and that Tobie is coming over to her place. Her relationship with Jackie is good they love each other and have raised a child together “Tobie” she gets sad when her and Tobie talks about Jackie because she has died of......

Words: 565 - Pages: 3

#### Birthday Cake

...For many people a birthday marks a day of festivities: inviting family and friends over for a celebration with a meal, presents, and the popular cutting of the birthday cake. However, I believe that Paul Goodman’s “Birthday Cake” depicts a painful and upsetting scene with the celebration of a birthday. Through his choice of words, images portrayed in the poem, and symbols, Goodman reveals a dark point of view on this special day: birthdays celebrate death rather than life for the old. The first clue of the painful reminder of death is the use of anaphora in the first stanza. Goodman repeats “too many” four times in three quick lines: when the candles on the icing are one two too many too many to blow out too many to count too many The first time the phrase is used, Goodman is referring to the candles on the cake—there are too many candles. Candles on a birthday cake represent the age of the birthday person; too many candles indicate old age. Goodman emphasizes the amount of candles with a pun by using the words two and too, successively. The speaker is reminded of his old age with the excessive amount of candles. Anaphora continues when Goodman reiterates, “now isn’t it time,” in the first and last line of the stanza. This first stanza ends in a rhetorical question,......

Words: 1012 - Pages: 5

#### A Birthday Remembered

...”A Birthday Remembered”(1980) Love is a very complicated and uncontrollable matter. It cannot be defined. And age, race and religion don’t matter when you are in love with another person. Love has no limit. It is unconditional. Love is not always a walk on the sunny side, but when it is, it is the most beautiful thing in the world. The American short story “A Birthday Remembered” is about how a family is handling the loss of a mother, a girlfriend, and an ex-wife. It is also about homosexuality and about all kinds of love. The story portrayed in a middle class society where money is not easy to find, especially for Roger who is the father and late husband in the story. We start in Ellen Simms house, which are the narrator of the story. It is Ellen’s birthday, which also happens to fall on Valentine’s Day. Ellen is a woman in her late thirties. She is a very sufficient woman, who lives by herself in a small house. She has been living ever since her partner Jackie died, which was life ruining for her. Ellen tries to still be in touch with Jackie’s child. She knows that Jackie wanted her to take care of Tobie. Ellen is a nice and non-judgmental. “Come in and take off your coats. Would you like some hot cocoa to warm you up? I Know its cold outside” She tends to cry very easily, and tends to drink when she is grieving. She misses her partner. Ellen is a lesbian, and Roger is not happy about it. And very sad for Ellen, because she cannot bury her loved one because......

Words: 646 - Pages: 3

#### 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......

Words: 572 - Pages: 3

#### A Happy Birthday

...“Happy Birthday!”, what does that even mean? A ‘wish’. A wish? A wish for my birthday girdled in happiness. With no trace of sorrow and misery and all things making life what it truly is. Sometimes I feel that the wish had enough, and rebelled to be an order. A commandment for happiness. Required. Or demanded, of course. And what’s with happiness anyhow? Is happiness really the one truth? Can one not wrap themselves up with unhappiness on their birthday? The lone thing which they know can keep them warm and safe. Can one not feel lonely?, or grief?. Can one not celebrate sorrow?, heartache? And anguish? No. It is my birthday. I must be happy. I must shake off my depression and attempt to enjoy and revel in the astronomical splendor of the day; or at the very least, pretend to. And what is it, which I’m being asked to celebrate? What is so exceptional about a birthday? A day I cannot even remember. A day with no memorable significance. A day of ‘what’ and ‘how’, but no ‘why’. A day, stripped of its conventional festivity, holds no sense, and no memory which I can take to my grave. What should I celebrate then? Is it the celebration of the culmination of nine months, and more, of struggle which my parents braved?; the sacrifices they made and the pain my mother endured?; the ‘happiness’ which ‘they’ felt when they heard me cry for the first time?; The only time my tears brought a smile to their face. Then why am ‘I’ being celebrated? My adolescent......

Words: 445 - Pages: 2

#### A Birthday Remembered

...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 since Jackie walked out on Roger and came to live with her.’’(I 10-11)  The story must have taken place before the 1970’s. Because it was after the 70’s, that homosexuality got accepted.   It doesn’t seem like Ellen wants everyone to know about her relationship with Jackie, which can be seen as it maybe wasn’t......

Words: 637 - Pages: 3

#### 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