How to Use the Complement Rule in Statistics

In statistics, the complement rule is a theorem that provides a connection between the probability of an event and the probability of the complement of the event in such a way that if we know one of these probabilities, then we automatically know the other one. The complement rule comes in handy when we calculate certain probabilities. Many times the probability of an event is messy or complicated to compute, whereas the probability of its complement is much simpler. Before we see how the complement rule is used, we will define specifically what this rule is. We begin with a bit of notation.  The complement of the event  A, consisting of all elements in the  sample space  S  that are not elements of the set  A, is denoted by  AC. Statement of the Complement Rule The complement rule is stated as the sum of the probability of an event and the probability of its complement is equal to 1, as expressed by the following equation: P(AC) 1 – P(A) The following example will show how to use the complement rule. It will become evident that this theorem will both speed up and simplify probability calculations. Probability Without the Complement Rule Suppose that we flip eight fair coins — what is the probability that we have at least one head showing? One way to figure this out is to calculate the following probabilities. The denominator of each is explained by the fact that there are 28 256 outcomes, each of them equally likely. All of the following us a formula for combinations: The probability of flipping exactly one head is C(8,1)/256 8/256.The probability of flipping exactly two heads is C(8,2)/256 28/256.The probability of flipping exactly three heads is C(8,3)/256 56/256.The probability of flipping exactly four heads is C(8,4)/256 70/256.The probability of flipping exactly five heads is C(8,5)/256 56/256.The probability of flipping exactly six heads is C(8,6)/256 28/256.The probability of flipping exactly seven heads is C(8,7)/256 8/256.The probability of flipping exactly eight heads is C(8,8)/256 1/256. These are mutually exclusive events, so we sum the probabilities together using one the appropriate addition rule. This means that the probability that we have at least one head is 255 out of 256. Using the Complement Rule to Simplify Probability Problems We now calculate the same probability by using the complement rule. The complement of the event â€Å"We flip at least one head† is the event â€Å"There are no heads.† There is one way for this to occur, giving us the probability of 1/256. We use the complement rule and find that our desired probability is one minus one out of 256, which is equal to 255 out of 256. This example demonstrates not only the usefulness but also the power of the complement rule. Although there is nothing wrong with our original calculation, it was quite involved and required multiple steps. In contrast, when we used the complement rule for this problem there were not as many steps where calculations could go awry.​

A Comparison of Jack London and Stephen Crane. - 1481 Words

A Comparison of Jack London and Stephen Crane. Jack London and Stephen Crane were both well-known literary naturalists who died at relatively early ages. Despite having lived such a short life, Jack London lived a full life. He has achieved wide popularity abroad, with his work being translated into more than fifty languages, as well as having written fifty literary works in eighteen years. His stories in the naturalistic mode still continue to influence writers today. Stephen Crane was also an accomplished writer as well as a poet. He was among the first to express in writing a new way of looking at the world. Although Crane is also seen as one of the first American naturalistic writers, a Symbolist, and Imagist, the achievements which†¦show more content†¦Imagery is another important element which London uses to illustrate and emphasize his theme. In â€Å"To Build a Fire† Earl Labor sees the â€Å"mood and atmosphere, which is conveyed through repetitive imagery of cold and gloom and whiteness,† as bein g â€Å"the key to the story’s impact† (63). London does rely heavily on imagery to set the mood of the story, and in this way he draws a picture of the harsh environment that his character must endure. London uses imagery with such skill that the reader can almost feel the deadly cold of the environment and can almost hear the â€Å"sharp, explosive crackle† when the man’s spit would freeze in mid-air (119). Through the use of such vivid imagery, London guides the reader toward the realization of the story’s theme; the reader can visualize the man â€Å"losing in his battle with the frost† and therefor can picture man in his conflict with a cruel and uncaring universe (128). Symbolism is also an important element in â€Å"To Build a Fire†. David Mike Hamilton’s criticism, he says â€Å"the fire symbolizes life as does the white snow that falls at the beginning of the story.† He also views â€Å"the dark point in the midst of the stamped snow, foretelling the end of the fire, and thus of life† (2). I strongly agree with Hamilton’s criticism; â€Å"the dark point in the midst of the stamped snow† because it not only foretells the end of the fire but of the end of life itself. Stephen Crane uses numerous techniques to capture his reader’sShow MoreRelated Visions of The Primitive in Langston Hughes’s The Big Sea Essay examples6201 Words   |  25 Pagesas an alien territory, forbidden, dangerous yet compelling in its intensity. It is ironic that this very conception of Harlem was one of the key reasons why it was overrun and exploited by sensation-seeking white outsiders. 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How the Great Depression Changed American Economy free essay sample

The Great Depression was one of the biggest turning points in American history. Not only did it change the spending habits of U. S. citizens, but it changed how the government controlled the economy. New laws and acts were put into place that helped prevent something like the Great Depression from ever happening again. During the 1920s, America’s economy was extremely prosperous. Businesses were producing mass amounts of products, and because wages were high, consumers were buying them. However, the spending habits of the American people are what would lead to the economy’s downfall. People would invest most of their money in stocks, and spend the rest on items they didn’t really need. Not many people put too much of their income into savings, and those who did may have lost it anyways once the Great Depression started. The President during this time period was Calvin Coolidge. As shown in Document 1, Coolidge believed in a very â€Å"hands-off† policy when it came to government involvement in the economy. We will write a custom essay sample on How the Great Depression Changed American Economy or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page Coolidge’s administration sort of sat back and allowed the economy to prosper on its own. This did not help the country once the Great Depression hit. There were several factors that led to the Great Depression. However, the main cause of the economy’s failure was the stock market crash of 1929. As seen on the graph in Document 2, once the stock market crashed, the unemployment rate skyrocketed. Banks and businesses went out of business, and millions of people lost their jobs. Before the crash, below 2 million people were unemployed. Four years later, in 1933, over 12 million were unemployed. Banks completely failed, losing all of their money. This meant that any money people had saved in the bank had vanished. The President at the time of the stock market crash was Herbert Hoover. Hoover was oblivious to the crash of the economy and did not know how to deal with it. His administration did little to help the economy or restore the country. In Document 3, you can see that Franklin D. Roosevelt, while running for president, said that â€Å"the only efforts by the Hoover administration to cope with the distress of unemployment were to deny its existence. † Because of Hoover’s failure as President, and Roosevelt’s promises, Roosevelt was elected as the new President in 1932. He would go on to serve three terms due to his great success in leading the country. Once Roosevelt became President, the government’s response to the Great Depression changed. In Document 4, Roosevelt says in his First Inaugural Address that he understands the problems of the American people and can sympathize with them. Because he personally understood what they were going through, it made them think that he would be able to help the country. His administration took more control over the economy and through a long, slow process, it gradually improved. In the first 100 days of his presidency, he shut down all banks that clearly were not going to assist the economy. He gave â€Å"fireside chats† to the American citizens, and personally explained to them how he was going to improve the economy. What truly brought the United States out of the Great Depression was Roosevelt’s New Deal. He created many important programs that aimed at providing economic relief for workers and farmers and creating jobs for the unemployed. He also initiated a slate of reforms of the financial system that helped protect depositors’ accounts and regulate the stock market. In 1935, Roosevelt created a new wave of reforms known as the â€Å"Second New Deal. † This included the Social Security Act, which for the first time provided Americans with unemployment, disability, and pensions for old age. Congress also raised taxes on large corporations and wealthy individuals. While the acts Roosevelt enforced with the New Deal vastly improved the economy, many American citizens were weary of them. In Document 5, there is a cartoon portraying an artist’s view on the New Deal. He saw it more as a way for people to test out their new ideas on the American economy than as a way to firmly improve the economy. During the Great Depression, the government set many new laws in place regarding the stock market and the general economy to make sure this kind of disaster never occurred again. They also wanted to ensure that if it did, the American people would not lose all of their money and jobs. They wanted to make the economy safer and more stable. Thanks to Franklin Roosevelt and his administration, the government was successful.