Conditional Probability Notation and Examples

A straightforward example of conditional probability is the probability that a card drawn from a standard deck of cards is a king. There is a total of four kings out of 52 cards, and so the probability is simply 4/52. Related to this calculation is the following question: What is the probability that we draw a king given that we have already drawn a card from the deck and it is an ace? Here we consider the contents of the deck of cards. There are still four kings, but now there are only 51 cards in the deck. The probability of drawing a king given that an ace has already been drawn is 4/51. Conditional probability is defined to be the probability of an event given that another event has occurred. If we name these events A and B, then we can talk about the probability of A given B. We could also refer to the probability of A dependent upon B. Notation The notation for conditional probability varies from textbook to textbook. In all of the notations, the indication is that the probability we are referring to is dependent upon another event. One of the most common notations for the probability of A given B is P( A | B ). Another notation that is used is PB( A ). Formula There is a formula for conditional probability that connects this to the probability of A and B: P( A | B ) P( A ∠© B ) / P( B ) Essentially what this formula is saying is that to calculate the conditional probability of the event A given the event B, we change our sample space to consist of only the set B. In doing this, we don’t consider all of the event A, but only the part of A that is also contained in B. The set that we just described can be identified in more familiar terms as the intersection of A and B. We can use algebra to express the above formula in a different way: P( A ∠© B ) P( A | B ) P( B ) Example We will revisit the example we started with in light of this information. We want to know the probability of drawing a king given that an ace has already been drawn. Thus the event A is that we draw a king. Event B is that we draw an ace. The probability that both events happen and we draw an ace and then a king corresponds to P( A ∠© B ). The value of this probability is 12/2652. The probability of event B, that we draw an ace is 4/52. Thus we use the conditional probability formula and see that the probability of drawing a king given than an ace has been drawn is (16/2652) / (4/52) 4/51. Another Example For another example, we will look at the probability experiment where we roll two dice. A question that we could ask is, â€Å"What is the probability that we have rolled a three, given that we have rolled a sum of less than six?† Here the event A is that we have rolled a three, and the event B is that we have rolled a sum less than six. There are a total of 36 ways to roll two dice. Out of these 36 ways, we can roll a sum less than six in ten ways: 1 1 21 2 31 3 41 4 52 1 32 2 42 3 53 1 43 2 54 1 5 Independent Events There are some instances in which the conditional probability of A given the event B is equal to the probability of A. In this situation, we say that the events A and B are independent of one another. The above formula becomes: P( A | B ) P( A ) P( A ∠© B ) / P( B ), and we recover the formula that for independent events the probability of both A and B is found by multiplying the probabilities of each of these events: P( A ∠© B ) P( B ) P( A ) When two events are independent, this means that one event has no effect on the other. Flipping one coin and then another is an example of independent events. One coin flip has no effect on the other. Cautions Be very careful to identify which event depends upon the other. In general P( A | B) is not equal to P( B | A). That is the probability of A given the event B is not the same as the probability of B given the event A. In an example above we saw that in rolling two dice, the probability of rolling a three, given that we have rolled a sum of less than six was 4/10. On the other hand, what is the probability of rolling a sum less than six given that we have rolled a three? The probability of rolling a three and a sum less than six is 4/36. The probability of rolling at least one three is 11/36. So the conditional probability in this case is (4/36) / (11/36) 4/11.

Apple Inc. Is An American Worldwide Organization That Is

Apple Inc. Is an American worldwide organization that is exploring by making and selling personal computers, computer software, and many more electronics devices? It is the well-known hardware products, they are the Mac line of computers, the iPod media player, the iPhone smartphone, and the iPad tablet computer. They also have their online services recognized, such as ITunes store, app store and ICloud. Apple Inc. products are very classy, original and they offer as well waterproof products. However, they still remain competitive in today s belligerent market, they must continue to devote in their organization through research or exploration development, marketing and advertising. Steve Jobs based on the way he carried his life and the†¦show more content†¦He also organized the development of the IMac, a new line of affordable home desktops, which debuted in August 1998 to rave reviews. Apple Inc. provide the perfect Scheme and efficacy. For example, the desktop computer is a portion of the screen, it is not a detached package with supports or wires; the iPhone doesn’t have a lot of buttons and it textures perfectly when holding. Beside all advantages Apple Inc. have, one of the principal benefits of Apple as a company is the boundaries that it guidelines above its merchandise. Moreover, the pricing tactic does not break within the market, and we will continuously discovery Apple selling or vending at the same price in different selling stores. Apple Inc. products are create with nice profile, and they are not too difficult to use, but at the end people get used to them and love it. It also almost intuitive. As per Mark Rogowsky: Apple s decision to remove the headphone jack from the iPhone 7 has generated its share of controversy to be sure. But it s also an illustration of a singular vision in the company: find a pain point, design a fix to eliminate the pain, and turn a profit doing it, that is a good strength for the company. In brief, App le have a lot of strengths and based on their quality, people always buying their products. The greatest part of Apple, and the purpose for its fantasticShow MoreRelatedCase Study : Apple Inc.1446 Words   |  6 PagesSummary Apple Inc. was built up on 1st April 1976 around then the organization was known as Apple PC, Inc. Apple almost has 35,000 representatives overall and its business recording about US$42.90 billion by 26th September 2009. It arrived in a magazine named Fortune that Apple had been most refreshing organization in the United States in 2008 and worldwide in 2009. Apple fundamentally offers tablets, iPods, and mobiles and so forth here we are set to focus on their portable PCs. 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BELARDOStudent no.: 10 -11077Submitted to: Professor MedinaDate: December 14, 2012 | | TABLE OF CONTENTSINTRODUCTION†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦3BODY/ RESEARCH†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦...6REFERENCES†¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦Ã¢â‚¬ ¦...10 | | INTRODUCTION Apple Inc. Is an American multinational electronics and software company established by Steven Jobs and Stephen Wozniak on April 1, 1976, in Cupertino, California. Apple Inc. Has becomeRead MoreApple Inc. Steve Jobs1431 Words   |  6 Pages APPLE INC. Steve Jobs Apple Inc. 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Someone Who Has Inspired Me free essay sample

My family is very important to me; they have encouraged me through good times and supported me through bad. My family has shaped who I am today and has always been there for me. I have seven aunts, nine uncles, thirteen cousins and a brother named William and a sister named Shauney. Out of my thirty one family members there is one I particularly look up to beyond all the rest, my Aunt Erin. My Aunt Erin has been by my side since the day I was born, she has supported me through numerous school activities such as cross country running races and outdoor track meets, but has also been by my side through many road races, and included me in marathons she has done as well. She is not only my aunt, god mother, running buddy and friend, but a role model to me as well and I look up to her eminently. We will write a custom essay sample on Someone Who Has Inspired Me or any similar topic specifically for you Do Not WasteYour Time HIRE WRITER Only 13.90 / page My aunt has inspired me to aim for my goals, work hard at it, and achieve. She graduated CVU High School; College bound and attended Johnson state college with an aspiration to become a special educator. There at Johnson she attended school for four years, worked exceedingly hard, was placed in the hall of fame with a soccer award, and graduated. She now works at Hinesburg community school as a special educator for k-2nd graders, and achieved her goal. When you’re young you are told that anything is possible if you set your mind to it, or at least that’s what my mom would always tell me. But as you mature and become more sophisticated, you realize that you are in fact limited to specific opportunities based on the direction you chose to take, and the decisions you make. I personally believe that not everyone can be successful in achieving their goals, but that everyone is given the opportunity and for me it’s how hard I chose to work on achieving my goals, and how important it is to me to succeed. My aunt encourages me, and I will strive to be like her. My goal is to become a dental assistant, it has been a passion of mine for quite a while and although my interests in careers has switched around a lot over the years, I have finally found the career path I choose to take. My goal has also always been to find a career in which I can help people no matter where I end up in the work force and I find dental assisting to be rather fitting. I admire the people who partake in this career, and I look forward to foreshadowing one of the dental assistants in my office within the next month or so, to get a greater idea as to what they do. When given the opportunity I have no doubt that I can achieve my goal, because I am ambitious, determined, dedicated, distinguished and when I set a goal for myself, I work hard to achieve it.