Various Factors Involved in Employee Performance Essay

Various Factors Involved in Employee Performance - Essay Example Also, in today's business community, there is far more focus on customer satisfaction. Performance appraisal (PA) is an important part of any organization, but according to human resources consultant John Drake, it seldom improves employee performance and sometimes even has a negative effect (1). Within any company, documentation is necessary for legal purposes, for feedback, corporate planning, employee development, and salary administration. Structure is needed in a PA plan, but it must be set up in a way that will help an employee understand and try to meet company goals and will best reflect the company's identity in the marketplace (Introduction). In the management process, it is necessary for a company to develop an initial plan for performance appraisal by meeting with a new employee and having a discussion that addresses key job responsibilities, a common understanding of company goals and objectives, the most important competencies needed, and an appropriate individual development plan (Grote 2.1). It is not enough just to pass out a manual, as has been the case in the past, and expect the employee to follow it. Without this initial discussion, further appraisal of the employee's attributes cannot be honestly determined. If an employee understands at the outset what is expected, his or her evaluation at different intervals will offer a more realistic appraisal of the employee's development. Once a performance appraisal method has been developed, the various factors that influence performance can be evaluated. With the current development of what might be called virtual organizations, it is employee relations, market relations and various hybrids of these two units that make up the overall performance factors for employees in today's global companies. Strict working hours and a single workplace have given way to a workforce that might be working from home or on the road as well as in the office itself. In the virtual marketplace, the physical employee becomes a combination of internal and external employees at different levels of training and expertise. In determining an appraisal method that takes all of this into consideration, it is necessary to recognize the factors that have not worked in the past and to replace them with positive factors based on a changing business arena. The boundary between organizations and the marketplace is changing as employees become more i nvolved in organizational structure. Labor agreements and internal company rulebooks are giving way to service contracts in companies that are beginning to depend on information technology (Huiskamp & Kluytmans). With the Internet bringing customers into the mix, changing expectations call for a more fluid method of appraisal based on individual performance within a flexible time frame, and in a global economy, changes are inevitable and ongoing. Factors Influencing Employee Performance From Negative to Positive: 1) Monetary incentives - According to Drake (14), "you cannot expect a performance appraisal to improve performance when it is also linked directly to rewards (salary or promotion)." Drake sets forth a situation in which a plant manager is limited in his ability to get a salary increase for an employee unless he rates the employee at least at "5" on a scale of 1-7, which means a rating of "outstanding." If the

Criminal Justice Process Essay Example for Free

Criminal Justice Process Essay Juveniles are not extended the right to a jury of their peers. What is the most significant reason why this right is not extended to juveniles? Please explain in detail why you chose that particular reason. I do not think that there is only one significant reason as to why juveniles are not extended the right to a jury of their peers. I think that one of the multiple reasons for this right not being extended is because a jury has to be able to be responsible enough to actually show up, they have to be able to convict or not convict based on evidence beyond a reasonable doubt and be able to disregard any evidence or statements when a judge asks them to. I do not think that most juveniles are mentally, nor emotionally equipped to do deal with the responsibilities of being a part of a jury. Most juveniles are still immature and when someone’s life is at stake and their freedom can be taken away I do not think that having a â€Å"child† decide their fate is appropriate. Why do you believe that there are differences between the adult justice system and the juvenile justice system? Please explain in detail why you believe as you do? I believe there are differences between the adult justice system and the juvenile justice system because of the differences in age, experiences, knowledge, and maturity. The juvenile justice system focuses on the rehabilitation of the juvenile, whereas the adult justice system’s goal is to punish and obtain retribution for the crime(s) committed. Juvenile offenders are given sentences that seek to rehabilitate rather than punish. Some of the treatment options offered could include counseling and placement in juvenile institutions that were also created to help restore them. Adult offenders are given harsh sentences. The sentences given can include a fine and or incarceration in jail or prison. All of the sentences given are solely based on punishing the offender. The adult court system is primarily concerned with the offender paying for the damage that they have done to society and the courts isn’t interested in rehabilitating or trying to change the behavior of the offender. After reviewing this entire building in CJ Interactive, describe ways that you can use this interactive tool to improve your learning of criminal justice concepts. Describe in detail the ways you can use this tool to further your criminal justice education, identifying at least three specific ways you would use this tool. After reviewing the entire building in CJ Interactive, I was able to see how this tool will help my learning of criminal justice concepts. I am a visual and auditory learner and both of my learning styles are incorporated in the CJ Interactive tool. I was able to get a better understanding of many of the topics that we have discussed in class thus far. For example through this too I was able to get a better understanding of the differences between the adult and juvenile court systems as well as how crime is defined and measured. I will be able to use this tool as another way of learning and understanding the criminal justice system by using the glossary to learn the terms used to explain the criminal justice system and its process. I will also be able to use this tool to learn and understand the criminal justice system by utilizing the different ways information is given is given in CJ Interactive for example there are 14 buildings located in this learning tool and each building represents a different topic in criminal justice and gives us students access to different learning modules, myths and issues, simulation activities, homework and review, and glossary terms associated with each particular topic. I can see myself utilizing all of these resources as a way to better understand the criminal justice system, my assignments that I have to complete in class, and for me to just use to gain as much knowledge as I can about the different topics in criminal justice throughout my college career at Colorado Technical University.

Determination of the End Point of the Acid Base Titration

Determination of the End Point of the Acid Base Titration Table of Contents (Jump to) Introduction Acids and Bases Properties of acid Strengths of Acids and Bases How to detect acid and Bases? pKa and Dissociation Equilibrium Equipment Procedure Results and conclusion Bibliography Introduction Acids and Bases Every liquid we see will probably have either basic or acidic properties. Water can be a base and acid, it depends on the reaction you add with water. It can be a base in some reaction and an acid in some reactions. Also water can react with itself to form bases and acids but it happens in small quantities so it will not change your experiments. 2H2O > H3O++ OH- The hydrogen ion was transferred to form Hydronium ion. The negative and positive ions in water are equal and cancel each other. Most of water we drink from the tap has others ions in it. Those ions in solution make something basic or acidic. For example, in our Bodies, there are small compounds called amino acids and in fruits there something called citric acid. According to Santà © Arrhenius, in 1887, he came up with new definitions of acids and bases. He said when we mix water to molecules , they break down and gives a hydrogen ion and at another times it gives hydroxide. In general, a hydrogen positive ion is released, the acidic solution increases. When a hydroxide ion is released, the solution become base For example HA +H2O H3O + + A Hydronium ion is formed and it is acid. That hydrogen ion is the reason it is called an acid. Chemists use the word dissociated to describe the breakup of a compound Properties of acid Acids taste sour Acids react strongly with metals (Zn + HCl) Strong Acids are dangerous and can burn your skin Bases Bases are ionic compounds that break apart to form a negatively charged hydroxide ion (OH-) in water. The strength of a base is determined by the concentration of Hydroxide ions (OH-). The greater of the concentration of OH ions the stronger the base. Example: NaOH in water NaOH Na+ + OH Strengths of Acids and Bases Strong Acids and Weak Acids: Strength of acid is related to ionization of acids in water. Some of the acids can ionize 100 % in water solutions; we call them strong acids. HCL are examples of strong acids.in other hand, some of the acids cannot ionize like strong acids. We call acids partially ionize in solutions weak acid. CH3COOH, HF, H2CO3 are examples of weak acid that partially ionize in solution Strong and Weak Bases: Bases ionize completely in solutions are called strong bases. NaOH and bases including OH- ion are strong bases. Bases that ionize partially in solutions are called weak bases. For example [ NH3] Ionization of Water: Water ionizes gives: H2O(l) ↔ H+(aq) + OH(aq) In pure water concentrations of H+ and OH ions are equal to each other and at 25 °, they have concentration 110-7 M. then concentration of ion in pure water is too low, it is a bad electric conductor. As in the case of pure water mediums having [H+] = [OH] concentration are called neutral mediums. In water solutions multiplication of [H+] and [OH] is constant and at 25 0C it is 110-14. This number is also called ionization constant of pure water. If concentration of [H+] ions equal [OH-]= 10 -7M, then solution is neutral. If concentration of [H+] ions > [OH-] or [H+] > 10 -7M and [OH-] -7 M, then solution is acidic. If concentration of [OH-] ions > [H+] or [H+] -7 M and [OH-] > 10-7 M, then solution is basic. How to detect acid and Bases? Scientists use something called pH scale to measure how basic or acidic the liquid is. Also there are many types of ions in a solution, pH focus on concentration of hydrogen ions and hydroxide ions. The scale measures values from 0 to 14. Distilled water is 7 in the middle. The strength of an acid or base in a solution is measured on a scale called a pH scale. Any pH number greater than 7 is considered a base and any pH number less than 7 is considered an acid. 0 is the strongest acid and 14 is the strongest base. The acid strength depends on the concentration of positive hydrogen ions in the solution. The greater and more hydrogen ions is the stronger acids likes Hydrochloric acid HCL and Sulphuric acid. pH=-log[H+] and pOH=-log[OH-] If 7>pH>0 acidic solution If 14>pH>7 basic solution If pH=7 neutral solution pKa and Dissociation Equilibrium 1. pH When acids is added to water, the pH scale decreases. The acidity of a solution is examined by the hydrogen ion concentration ([H+]), where pH provides a simple index for expressing the [H+] level., when pH is small which means that the smaller the number of pH , the stronger acid. pH=-log10[H] pKa and Dissociation Equilibrium Strong acid , which they are dissociate in solution, and weak acids that partially dissociate in solution. When dissociation of strong acid happens, it gives a proton In which make the solution more acidic, However, weak acids have a dissociated state (A-) and undissociated state (AH) that appears according to the following dissociation equilibrium equation. AH A + H+ . The definition of Ka is Ka= The brackets of the product to the brackets of the reactants pKa was introduced as an index to express the acidity of weak acids, where pKa is defined as follows. pKa= log10Ka Relation between Ka and Pka , it is inversely proportional so when ka is high which means storng acid which means pKa is low and vice versa Equipment Burette Beaker Magnetic stirrer Ph meter Acid and Bases Pure water Procedure Clean all equipment in order to get accurate conductivity Add some of NaOH into the receiving cup and then add slightly 1 ml of HCL and make sure you adding the receiving cup on the magnetic stirrer Repeat this steps to get the conductivity from volume 0 ml to 17 ml Get titration curve ,the differential curve and the end point For CH3COOH + NaOH We will make same steps and record the conductivity pH Results and conclusion NaOH + HCL Results of volume of HCL and the conductivity Ml pH is clear that end point occurs at 10 ml of HCL which pH drops to 6.34 Titration curve Differential curve Ch3COOH +Naoh It is clear that end point at 10 ml of Naoh the end point occur , the ph difference is very big Titration curve Differential curve Bibliography http://www.chemistrytutorials.org/content/acids-and-bases/ph-poh-and-ionization-of-water/58-acids-and-bases-cheat-sheet http://www.sciencebuddies.org/science-fair-projects/project_ideas/Chem_AcidsBasespHScale.shtml http://www.elmhurst.edu/~chm/vchembook/184ph.html http://lrs.ed.uiuc.edu/students/erlinger/water/background/ph.html http://www.humboldtmfg.com/graduated_glass_beaker.html http://www.shimadzu.com/an/hplc/support/lib/lctalk/29/29intro.html https://www.boundless.com/chemistry/acids-and-bases/strength-of-acids/the-acid-dissociation-constant/ http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Acid_dissociation_constant.html

Trinculo and Stephano of Shakespeares Tempest :: Tempest essays

Trinculo and Stephano of Shakespeare's Tempest Trinculo and Stephano though not major characters in William Shakespeare's The Tempest, serve a large role in the story itself. They mainly serve as the story's comic relief and they also contribute to demonstrating to the audience how evil has no boundaries. Much of the play revolves around Prospero's contempt for everyone who betrayed him, and Prospero forces the conspirators to a remote island. Trinculo and Stephano had nothing to do with the plot against Prospero, but end up being dragged along with the conspirators. Their parts were small but were probably the most interesting in the story. Trinculo and Stephano were primarily used for comic relief. Comic relief is very important because the story must be able to keep the audience interested. What better to make someone laugh than a pair of drunk servants. During the duration of the story their drunkenness causes them to do things that normally they wouldn't do. They blindly attempt to take on Prospero, a powerful sorcerer and scheme how to defeat and kill him. Who in their sober mind take on an all powerful sorcerer? This is quite amusing because it shows us how incredibly foolish we act when we are drunk. Of course their attempt to take on Prospero proves to be futile, instead they play dress up with his cloaks and when Prospero shows up, Stephano and Trinculo run for their lives and leave Calaban behind carrying the clothes they attempted to steal. Trinculo and Stephano were also quite amusing by being drunk throughout the entire story, they even stated that they wouldn't drink anything else until the wine ran out. "Tell not me. When the butt is out, we will drink a drop of water, not a drop before."(Tempest 288) Trinculo and Stephano also contribute to the play the idea that evil in men shows no boundaries. While Antonio and Ferdinand are making a plot to kill the King, Alonso, for power, Trinculo and Stephano are doing the same towards Prospero. They were enchanted by the story told by Calaban that they would become in charge of the island once Prospero was disposed of. Another display of their lack to purity comes in the speech by Trinculo on finding the resting Calaban. Trinculo stated that if he were back home that he would have people pay to see Calaban. "Were I in England now, as I once was, and had but this fish painted, not a holiday Trinculo and Stephano of Shakespeare's Tempest :: Tempest essays Trinculo and Stephano of Shakespeare's Tempest Trinculo and Stephano though not major characters in William Shakespeare's The Tempest, serve a large role in the story itself. They mainly serve as the story's comic relief and they also contribute to demonstrating to the audience how evil has no boundaries. Much of the play revolves around Prospero's contempt for everyone who betrayed him, and Prospero forces the conspirators to a remote island. Trinculo and Stephano had nothing to do with the plot against Prospero, but end up being dragged along with the conspirators. Their parts were small but were probably the most interesting in the story. Trinculo and Stephano were primarily used for comic relief. Comic relief is very important because the story must be able to keep the audience interested. What better to make someone laugh than a pair of drunk servants. During the duration of the story their drunkenness causes them to do things that normally they wouldn't do. They blindly attempt to take on Prospero, a powerful sorcerer and scheme how to defeat and kill him. Who in their sober mind take on an all powerful sorcerer? This is quite amusing because it shows us how incredibly foolish we act when we are drunk. Of course their attempt to take on Prospero proves to be futile, instead they play dress up with his cloaks and when Prospero shows up, Stephano and Trinculo run for their lives and leave Calaban behind carrying the clothes they attempted to steal. Trinculo and Stephano were also quite amusing by being drunk throughout the entire story, they even stated that they wouldn't drink anything else until the wine ran out. "Tell not me. When the butt is out, we will drink a drop of water, not a drop before."(Tempest 288) Trinculo and Stephano also contribute to the play the idea that evil in men shows no boundaries. While Antonio and Ferdinand are making a plot to kill the King, Alonso, for power, Trinculo and Stephano are doing the same towards Prospero. They were enchanted by the story told by Calaban that they would become in charge of the island once Prospero was disposed of. Another display of their lack to purity comes in the speech by Trinculo on finding the resting Calaban. Trinculo stated that if he were back home that he would have people pay to see Calaban. "Were I in England now, as I once was, and had but this fish painted, not a holiday

Ben and Jerry’s Entering into the Japanese Market

Ben and Jerry's Entering into the Japanese Market sy Ihart2dance19 Ben ; Jerrys Homemade, Inc. produces super premium ice cream, frozen yogurt, and ice cream novelties in rich and original flavors. The company sells its unique offerings In grocery stores, restaurants, and franchised Ice cream shops, and it holds about one-third of the market for its products. This global company began with only a $12,000 Investment to open Ben ; Jerrys Homemade Ice cream scoop shop In a renovated gas station in downtown Burlington, Vermont, on May 5th, 1978.From one mall shop In downtown Burlington, the company had grown to Include a chain of nearly 100 franchised shops, and a line of products sold in stores across the country. As one of the leading superpremium ice cream (greater richness and density than other kinds of Ice-cream and Is therefore sold at a relatively high price) manufactures, Ben ; Jerrys has to continually expand and develop to compete with other leading brands. The united States I s one of the largest exporting nations as well.The united States sells products to other countries because no country can roduce all of the products the people want. In 1994, den ;Jerrys starting considering advancing into the Japan ice cream market, the second largest ice cream market in the world with sales of approximately $4,5 billion. According to the survey conducted by â€Å"What Japan Thinks,† nearly 2 out of 5 Japanese eat ice cream every week. However, Japan is a great distance from the united States and it would be complicated to distribute the Items to Japan.Japan's barriers to Imports from foreign countries were high and Ben ; Jerrys were entering the Japanese ice cream market 0years atter Its competitors, such as Haagen-Dazs. According to the survey by â€Å"What Japan Thinks,† the biggest factor in ice cream purchase is by flavor and taste. The Japanese consumers demand high-quality products with different flavors. The demands of the Japanese coincide di rectly with the product mission statement of Ben ; Jerrys which is â€Å"to make, distribute and sell the finest quality all natural ice cream and euphoric concoctions with a continued commitment to Incorporating wholesome, natural ingredients. So based on the quality and flavors of Ben & Jerrys, the ompany doesn't have to change their recipes or ingredients to be popular In the Japanese ice cream market. However, In Japan ice cream is considered a snack more so than a dessert, so to be user- friendly to the Japanese, Ben ; Jerrys should package their Ice cream In personal cups as well as their point sized package. Additionally, the Japanese are very clean and conscience of sanitation, so having Individual serving would be more appealing to the Japanese people.According to â€Å"What Japan Thinks,† the most popular purchase of ice cream is a single-serving cup ot ice cream. When It comes to perishable goods, supermarkets seem to be much stricter In Japan than In the west abo ut moving on stock before it gets old. It Is very important for a product to have a good reputation, especially in Japan, and if a product Isnt good quality no one will buy the product. Ben ; Jerrys should make sure that their product's are being monitored, and if the ice cream is close to perishing, they should make sure It gets thrown out, or then their reputation can be ruined In a 1 Ofa minute. nen Ben & Jerrys aec10e now tney wlll Introduce tnelr product to Japan, hey have to take into account the sociocultural forces and cultural differences between America and Japan. Although shipping to Japan is not the easiest task, Ben & Jerrys is an established corporate company who has been shipping ice cream to the West Coast and to Europe in freezer containers. Ben & Jerrys needs to create an efficient supply chain, the sequence of linked activities that must be performed by various organizations to move goods from the sources of raw materials to ultimate consumers, so the company can then ship out their products smoothly.The company hen has to find the best approach to their physical distribution, or logistics. Bringing their products to Japan would require detailed and structured outbound logistics involving managing the flow of finished products and information to business buyers and ultimate consumers. Ben & Jerrys then has to choose the right transportation mode. Because Japan is over seas from their Vermont factory, the only 2 options would be water transportation, which is inexpensive but slow (about 3 weeks) or by air, which is fast but expensive.Although Japan has barriers to foreign imports, in 948 the General Agreement of Tariffs and Trade (GATT) was formed, which was an international forum for negotiating reductions in trade restrictions. The World Trade Organization (WTO) was also established to assume the task of mediating trade disputes among nations. Japan is part of the WTO, Joining on September 10th, 1955. This will make it easier for Ben & Jerr ys to advance in Japan's foreign market because there is a global mediation center. Also, there are expectations of falling tariffs on dairy products, which would be a desirable feature in selling in Japan.Even though Haagen-Dazs had already been selling their superpremium ice cream in Japan's market, now Ben & Jerrys doesn't have to educate the Japanese market about superpremium ice cream. Haagen-Dazs's sales in Japan were about $300 million, proving there is a large Japanese ice cream market and superpremium ice cream is desirable in the country. There are many advantages and disadvantages for Ben & Jerrys to penetrate the Japanese market by relying on 7-Eleven, an international chain of convenience stores, to distribute their superpremium ice cream.If Ben & Jerrys sold directly to 7-Eleven creating a Joint venture or a strategic alliance, they would create a long-term partnership between two companies to undertake a major project and help each company build competitive market adv antage. Because Ben & Jerrys have expanded all over the world it is a multinational corporation. If Ben & Jerrys could sell directly to 7-Eleven, it would eliminate the distribution costs. However, there would be a power struggle between the 2 major companies.If Ben & jerry's agrees to an exclusive agreement with the massive convenience store chain, 7- Eleven would have the upper hand. Another advantage of entering the market through 7-Eleven is the immediate placement of Ben ; Jerrys in over 7000 convenience stores in Japan, giving Ben ; Jerrys an instant access to the market on a large scale. Yet, by doing this, Ben ; Jerrys might not be able to build their own brand name and an issue with 7-Eleven would leave Ben ; Jerrys without their own position in the Japanese market.Also, 7-Eleven insisted that Ben ; Jerrys ice cream be packaged in personal cups as opposed to the pint size, due to the cultural view of ice cream in Japan. This would require $2 million in equipment and differe nt methods in packaging the ice cream, because Ben ; Jerrys would have to comply wltn tnese cnanges. I ne ‘-Eleven approacn to Just-ln-tlme Inventory procedures would make delivery reliability key and costs would have to be minimized. Because the Japanese production is unique, Ben & Jerrys would have to be careful to not mix up the Japanese label with the regular label.A disadvantage of relying on 7-Eleven is the asset specific investment in production equipment. Due to these changes, there would be complex logistics and production planning. Also, the pricing and profit distributions are unclear. The only clear thing was that Ben & Jerrys would be shipping from their Vermont factory. Entering the market with 7-Eleven would allow Ben & Jerrys to have control of their brand, although 7-Eleven would have a dominant position. Ben & Jerrys would have to rely on 7-Eleven promoting the brand, which 7- Eleven wasn't promising.A major advantage is that 7-Eleven is an established corpor ation, so 7-Eleven has high-level executive involvement and an efficient supply chain. Ben ; Jerrys would increase sales through convenience stores and would ccess the market on a large scale easily. Ken Yamada was also interested in acting as a licensee for Ben ; Jerrys in Japan, overseeing marketing and distribution of its products there. Yamada would be the marketing intermediary for Ben ; Jerrys, being the independent firm which will assist in the flow of goods and services from producers to end-users.Yamada would be a good candidate because he was a well- recommended third-generation Japanese-American, so he knew the culture and how to integrate American and Japanese cultures. He also was already running the Domino's Pizza franchise in Japan. The Domino's franchise in Japan was very successful, and Domino's already delivered ice cream cups, so they had the resources to deliver Ben & Jerrys. However, part of Yamada's agreement was that he would have exclusive rights to the entir e Japanese market.This would mean that Yamada would have full control of branding and marketing efforts, making Ben ; Jerrys fully dependent on the efforts of Yamada. He would have full control of the marketing and sales in Japan. Yamada would introduce Ben ; Jerrys to the Japanese market from he initial steps to the large picture; starting with positioning the brand, formulating and strategically orchestrating the initial launch, and concentrating on the best marketing and distribution strategy for the long-term positioning of Ben ; Jerrys in Japan.By using Yamada to introduce Ben ; Jerrys in the Japanese market, Yamada would earn royalty on all sales, but he would have full control of the Japanese market. This would give Ben ; Jerrys instant expertise in a foreign market and because Yamada was already running Domino's, there was a simple entry strategy and an ongoing marketing management. Yamada was very valuable to the ice cream company. He knew frozen foods, he had an entreprene urial spirit and marketing sa'. n. y.However, because Yamada would be investing his time in a marketing campaign only after reaching an agreement with Ben ; Jerrys, there was no specific plan available for consideration, and Yamada would have full control and the right to change any plan. Yamada has good market knowledge and the managerial requirements, making it less demanding for Ben ; Jerrys. However, he has no specific business plan and no brand control. Although Ben ; Jerrys managers believe the ompany should delay entering the Japanese market because of economic problems, I think Ben ; Jerrys should enter the Japanese market.Japan is the second largest ice cream market globally, with sweet growth rates. Japan has high profit margins. Japan nas a nlgn aemana Tor super premium Ice cream. Inere Is also a aecllnlng aomestlc growth rates and market shares in Japan. Also, Ben ; Jerrys has excess capacity in the United States factory. Japan has the second largest ice cream market in the world with sales of approximately $4. 5 billion, proving that Ben ; Jerrys would be very successful entering the Japanese market.

Probability Questions on ACT Math Strategies and Practice

Probability Questions on ACT Math Strategies and Practice SAT / ACT Prep Online Guides and Tips What is the probability that you’ll toss a coin and get heads? What about twice in a row? Three times? Probability questions ask you determine the likelihood that an event or any number of events is to occur, and the more you practice, the better your odds will be at mastering these types of questions on the ACT (see what we did there?). This will be your complete guide to probability on the ACT- how probability works, the different types of probability questions you’ll see on the test, and the steps you’ll need to take to solve them. What Does Probability Mean? $\Probability = {\desired \outcome}/{\all \possible \outcomes}$ On the ACT, probability questions can be framed in several different ways. You may be asked to find the â€Å"probability† that an event will occur, the â€Å"chances,† the â€Å"odds,† or the â€Å"likelihood.† But no matter how you see it written on the test, these are all ways of asking for the same thing. The way we represent the probability of an event (or events) is to express, as a fraction, how often that event occurs over the total number of possible outcomes. So if we use our example from above- †What are the odds that you’ll flip a coin and get heads?†- the odds will be: ${\desired \outcome}/{\all \possible \outcomes}$ $1/2$ In this one throw, there is one possible chance of getting heads. This means our denominator is 1. There are also two possible outcomes total (heads or tails), which means that our denominator will be 2. Now let’s take a look at another example: Mara is stringing a necklace and she selects each bead at random from a basket of beads. If there are currently 5, yellow beads, 10 red beads, 15 green beads, and 20 blue beads in the basket, what are the chances that she will select a red bead next? ${\desired \outcome}/{\all \possible \outcomes}$ There are 10 red beads, which is our desired outcome. This means 10 is our numerator. There are also a total of $5 \yellow \beads + 10 \red \beads + 15 \green \beads + 20 \blue \beads = 50 \total \beads$ in the basket. This is our denominator, as it represents all the outcomes possible. When we put these together, our probability is: $10/50$ $1/5$ The chances that Mara will select a red bead are 1 in 5 or $1/5$. Now what if we framed our desired outcome as a negative? What are the odds that Mara will NOT select a green bead? In order to find a negative probability, we must subtract out the chances that Mara will draw a green bead. (We could also think of this as finding the desired outcome of her selecting a yellow bead, a red bead or a blue bead, which we will cover in more detail in the next section.) There are only yellow, red, green, and blue beads, so we can add up our odds of yellow, red and blue beads, excluding the green. There are 5 yellow beads, 10 red beads and 20 blue beads, so we can put those together to get our numerator. $5 + 10 + 20 = 35$ And there are still $5 + 10 + 15 + 20 = 50$ beads total for our denominator. So what are the odds that Mara will NOT select a green bead? $35/50$ $7/10$ The odds are 7 in 10 ($7/10$) that Mara will draw any color bead except green. Expressing Probabilities As you can see, probabilities are expressed as fractions. This means that an event that will always and absolutely occur will have a probability of $1/1$ or 1. On the other hand, an impossible event will have a probability of $0/x$ or 0. You can also think about probabilities as percentages. If the odds are $4/52$ that you’ll draw an ace from a deck of cards, it’s the same as saying that there is a 7.69% chance that you will draw an ace. Why? Because $4 à · 52 = 0.0769$, and $0.0769 * 100 = 7.69%$. The possibilities are (not quite) endless. Either/Or Probability ${\probability \of \either \event = [{\outcome A}/{\total \number \of \outcomes}] + [{\outcome B}/{\total \number \of \outcomes}]$ (Special note: this is called a â€Å"non-overlapping† probability. In this case, it is impossible for the two (or more) events to both happen at the same time. There is such a thing as an either/or probability for overlapping events, but you will never be asked to do this on the ACT, so we have not included it in this guide.) An either/or probability increases the odds that our desired outcome will happen because we do not care which of the two events happen, only that one of them does. To solve this kind of problem, we must therefore add the probability of each individual event. Their sum will become the probability of either event happening. So let’s look again at our earlier example with Mara and her beads. Instead of asking the odds of Mara selecting only a red bead, what are the odds that Mara will select either a red bead or a green bead if she has 5 yellow beads, 10 red beads, 15 green beads, and 20 blue beads in the basket? We have increased our odds, since it doesn’t matter whether or not the bead is green or red, so long as the bead we select is NOT blue or yellow (essentially, we are doing another version of our earlier negative problem- †what are the odds that a particular event will NOT happen?†) This means we can add the probabilities of our individual events together in order to find their combined probability. So let us find the probability of her drawing a red bead: $10/(5 + 10 + 15 + 20)$ $10/50$ And let us find the probability of her drawing a green bead: $15/(5 + 10 + 15 + 25)$ $15/50$ So, if we put the two probabilities together, we’ll have: $10/50 + 15/50$ $25/50$ $1/2$ Because this problem involves the odds of two events with the same total number of outcomes (there are 50 total possible beads to choose from each time), we could also simply add our two desired outcomes together over the total number of outcomes. So: $(10 + 15)/(5 + 10 + 15 + 20)$ $25/50$ $1/2$ Either way, the odds of Mara drawing either a red bead or a green bead are 1 in 2, or $1/2$ (50%). What are the odds that we go this way or that way? Combined Probability $\Combined \probability = [{\outcome A}/{\total \number \of \outcomes}] * [{\outcome B}/{\total \number \of \outcomes}]$ "What are the odds of two or more events both/all happening?" This kind of probability question is called a combined probability and there is a good chance you’ll see a question of this type in the later half of the ACT math section. Note that a combined probability question is distinctly different from an either/or probability question. An â€Å"either/or† question asks whether or not one of the multiple events occurs (no matter which event is was). A â€Å"both/and† question requires that multiple events all occur. To find the probability of an â€Å"either/or† question, we must add our probabilities. To find the probability of a combined probability question, we must multiply our probabilities. A good way to remember this is to remember that a combined probability question will ultimately have a lower probability than the that of just one (or either) event occurring. The more events you need to happen, the less likely it will be that they all will. How likely is it that your first AND second coin tosses will BOTH be heads? Lower than the odds of just flipping heads once. On the other hand, an either/or probability question will have higher odds than the probability of just one of its events happening. You are combining forces to increase your odds of getting a desirable outcome. How likely is it that you’ll flip either heads or tails for each toss? 100%! What are the odds that Jenny will roll a pair of dice and get six on both? A die has six faces, so the odds of rolling any particular number is $1/6$. Because the question is asking us to find the odds of rolling two sixes (and nothing else), we must use our combined probability. So: $1/6 * 1/6 = 1/36$ There is a 1 in 36 chance that Jenny will roll a pair of dice and get two sixes. Combined probability questions mean that events cannot be separated. Typical ACT Probability Questions There are many different kinds of probabilities and probability questions (including overlapping, and conditional probabilities), but ACT probability questions use only the basic probabilities we have covered above. For most ACT probability questions, you will be asked to find either a straight probability or a probability ratio. You may also be asked to find or alter a new probability from an existing one. Now let us look at each type of problem. Simple Probability These kind of questions will always be word problems in which you are told a story and asked to find the probability of one or more events. This may be a straight probability, an either/or probability, or a combined probability. Simply use the understandings we learned above and you’ll be able to solve these kinds of questions without issue. We know that probability is ${\desired \outcome}/{\all \possible \outcomes}$. Our desired outcome is to get one of the five extra pieces, so our numerator will be 5. There are 750 puzzle pieces PLUS the extra five pieces in the box total, so our denominator will be: $750 + 5 = 755$ When we put them together, our final probability will be: $5/755$ Our final answer is D. Probability Ratio One way the ACT likes to spin probabilities and make them more complex is to present them as ratios or to ask you for your answer in a ratio. For a refresher on ratios, check out our guide to ACT fractions and ratios. For these types of questions, pay close attention to what the ratio represents so that you don’t end up solving the wrong question entirely. We are told that we must find the odds of an event as a ratio of $\in \the 25 - 35 \age \range: \not \in \the 25-35 \age \range$ (in other words, $\desired \outcome: \remaining \outcomes$). We are given the number of voters in terms of percentages, so we can translate the 42% of voters in the 25-35 age range as $42/100$. And if the 25-35 age category has a probability of $42/100$, then the remaining voters will have a probability of: ${100 - 42}/100$ $58/100$ Now, we can represent our ratio of $25-35 \voters: \all \other \voters$ as: $42:58$ Both numbers are divisible by 2, so we can reduce the ratio to: $21:29$ Our final answer is D. Altering a Probability Finally, it is quite common for the ACT to ask you to alter a probability. Usually, they will present you with an existing probability and then ask you to find the number to which you must increase the desired outcome(s) and the total number of outcomes in order to achieve a specific new probability. For example: Now, there are two ways to solve this kind of problem- using proportions or using the strategy of plugging in answers. Let’s look at both methods. Method 1- Proportions We are asked to find an additional number of red marbles that we must add to the total number of marbles in order to find a new probability. The current probability of selecting a red marble is: $12/32$ Now, we are adding a certain number of red marbles and only red marbles. This means that the number of red marbles increases by exactly the same amount that the total increases. We can therefore represent the new probability as: ${12 + x}/{32 + x}$ Now, we want this new probability to be equal to $3/5$, so let us set them up as a proportion. ${12 + x}/{32 + x} = 3/5$ And because this is a proportion, we can cross multiply. $(32 + x)(3) = (12 + x)(5)$ $96 + 3x = 60 + 5x$ Now solve for $x$. $36 = 2x$ $18 = x$ So we must add 18 red marbles in order to get a new probability of: ${12 + 18}/{32 + 18$ $30/50$ $3/5$ Our final answer is G, 18. Method 2- Plugging in answers The alternative to using proportions is to use PIA. We can simply add the answer options to the 12 red marbles in our numerator and the 32 marbles in our denominator and see which answer choice gives us a final ratio of $3/5$. Let us begin, as always, with the answer choice in the middle. Answer option H gives us 28, so let us try adding 28 to both the red marbles and the total number of marbles. ${12 + 28}/{32 + 28}$ $40/60$ $2/3$ This answer is a little bit too large. We can also see that the larger the number we add to both the numerator and the denominator, the larger our probability will be (you can test this by plugging in answer choice J or K- for K, if you add 40 to both 12 and 32, your final probability fraction will be $52/72$ = $13/18$, which is even larger than $2/3$.) This means that we can eliminate answer choices H, J, and K. Now let us try answer choice G. ${12 + 18}/{32 + 18}$ $30/50$ $3/5$ We have found our desired ratio. Our final answer is G, 18. As you can see, no matter which method you use, you can find the right solution. Somebody's gotta win, right? Well, you are more likely that to get struck by lightening (odds: 1.3 million to 1) and THEN fall from a 15 story building and survive (odds: 90 to 1), than you are to win the lottery (odds: 120 million to 1). How to Solve a Probability Question There are several ACT math strategies you must keep in mind when solving a probability question. First of all, you will know if you are being asked for a probability question on the ACT because, somewhere in the problem, it will ask you for the "probability of," the "chances of," or the "odds of" one or more events happening. Almost always, the ACT will use the word â€Å"probability,† but make sure to note that these words are all interchangeable. When you see those phrases, make sure to follow these steps: #1: Make sure you look carefully at exactly what the question is asking. It can be easy to make a mistake with probability ratios, or to mix up an either/or probability question with a both/and question. Make sure you always carefully examine the problem before you waste precious time trying to answer the wrong question. Kyle has been tossing a coin and recording the number of heads and tails results. So far, he has tossed the coin 5 times and gotten heads each time. What are the odds that he will get tails on his next coin toss? You may be tempted to think that our desired outcome (our numerator) is influenced by the number of times Kyle has already tossed the coin and the outcomes so far, but in all actuality, the probability that Kyle will get tails on his next toss is $1/2$. Why? Because each coin toss is independent of another coin toss. This means that this is a simple matter of determining our desired outcome over the number of total outcomes. There is one possibility of getting tails- numerator 1- and two possible options- heads or tails, denominator 2. So Kyle’s chances of getting tails on the next toss are 1 in 2. Now let’s look at a slightly different question. Kyle tossed the coin 5 times and got heads each time. What were the odds of this happening? Now we are being asked to find the probability of a both/and question, since we are being asked to identify the probability of multiple events all happening. (If it helps to picture, you can rephrase the question as: â€Å"What are the odds that BOTH his first coin tosses were heads? And What were the odds that BOTH his next tosses were heads?†, etc.) So if we use what we know about combined probabilities, we would be able to say: $1/2 * 1/2 * 1/2 * 1/2 * 1/2$ $1/32$ The odds are 1 in 32 (3.125%) that Kyle would have tossed heads five times in a row. #2: Think logically about when your odds will increase or decrease The odds of either two or more events occurring will be greater than the odds of one of the events alone. The odds of both (or multiple events) all occurring will be less than the odds of the odds of one of those events alone. Always take a moment to think about probability questions logically so that you don’t multiply when you should add, or vice versa. #3: Simplify the idea of a probability Once you get used to working with probabilities, you’ll find that probability questions are often just fancy ways of working with fractions and percentages. A probability ratio is the exact same thing as a question that simply asks you for a ratio. Just brush up on your fractions and ratiosif you find yourself intimidated for any reason. And always feel free to fall back on your PIA or PIN,as needed. These methods will sometimes take a little extra time, but they will always lead you to the right answer. The probability of drawing this hand is less than 0.0000004%, so I'm gonna go ahead and go all in. Test Your Knowledge Now it’s time to test what you’ve learned, using real ACT practice problems: 1) 2) 3) 4) Answers: F, E, D, B Answer Explanations: 1. This is another example of an altering probability question and, again, we have two choices when it comes to solving it. Let’s go through both the algebra/proportion method and PIA. Method 1- proportions. We know that we must increase the number of red marbles and only red marbles, so the amount of new marbles added to the set of red marbles and to the overall total of marbles will be the same. Our starting probability of red marbles is: $6/18$ So now we must increase each part of our fraction by the same amount and set it equal to the desired probability of $â…â€"$. ${6 + x}/{18 + x} = 3/5$ $(18 + x)(3) = (6 + x)(5)$ $54 + 3x = 30 + 5x$ $24 = 2x$ $12 = x$ So we must increase our red marbles (and, consequently, the total number of marbles) by 12 in order to get a probability of $â…â€"$ of selecting a red marble. To double-check this, we can plug the number back into our probability. ${6 + 12}/{18 + 12}$ $18/30$ $3/5$ We have successfully found our answer! Our final answer is F, 12. Method 2- PIA The alternative method is to use plugging in answers. We will simply plug in our answer choices to increase our red marbles (and our total number of marbles) and see which answer choice results in a probability of $3/5$. Let us start with answer choice H, 18. ${6 + 18}/{18 + 18}$ $24/36$ $2/3$ This probability is too large and any larger numbers will only get us larger probabilities. This means we can eliminate answer choices H, J, and K. Now, let us try answer choice G, 16. ${6 + 16}/{18 + 16}$ $22/34$ $11/17$ This probability is still slightly too large. By process of elimination, our answer must be F, but let us test it to be sure. ${6 + 12}/{18 + 12}$ $18/30$ $3/5$ Success! We have found our right answer. Our final answer is, again, F, 12. 2. Because Elliott must answer all the questions correctly, this means that this is a combination probability question. We are told that he answers each question at random, and all the questions have 3 answer options, which means that answering one question correctly has a probability of: $1/3$ And, since this is a combination problem, answering ALL 4 questions correctly will be: $1/3 * 1/3 * 1/3 * 1/3$ $1/81$ Our final answer is E, $1/81$ 3. We have a total of 150 people and 67 of them have type A blood, while 6 of them have type AB. This means that type A blood has a probability of: $67/150$ And type AB blood has a probability of: $6/150$ Now we can add these probabilities together. $67/150 + 6/150 = 73/150$ Our final answer is D, $73/150$ 4. Here, we have another probability question made more complicated by the use of ratios. Again, if you need a refresher on ratios, check out our guide to ACT fractions and ratios. First, we must find the actual number of 10th and 11th graders. We are told that the 10th graders have a ratio of 86:255 to the school population and the 11th graders have a ratio of 18:51 to the total student population. We must first set these ratios to an equal number of total students in order to determine the number of students in each class. We can see that the 11th graders have a reduced ratio, so we must multiply each side of the ratio by the same amount in order to equal the total number of students as the 10th graders’ ratio (255). Luckily for us, $255/51 = 5$. This is a nice, round number to work with. Now, we must multiply the 11th grade ratio by 5 on each side to even out the playing field. $18(5):51(5)$ $90:255$ We are assuming for now that there are 255 students total (there may be $255 *2$ or $255 * 3$, and so forth, but this will not affect our final outcome; all that matters is that we choose a total number of students that is equal for all grades/ratios.) So there are 86 10th graders, 90 11th graders, and the remaining students are 12th graders. Knowing that there are 255 students total, we can find the number of 12th graders by saying: $255 - 86 - 90 = 79$ There are 79 12th graders. This means that the probability of selecting a 10th, 11th, or 12th grader at random is: $86/255$, $90/255$, $79/255$, respectively. The odds are higher that the lottery will select an 11th grader, as the numerator for 11th graders is larger than that of the others. Our final answer is B, 11th graders. You have successfully completed your probability questions! You're free! The Take Aways The more you practice working with probabilities, the easier they will become. Although it can take some time to learn how to properly differentiate between the different types of probability questions, most ACT probability questions are fairly straightforward. Understand that probabilities are simply fractional relationships of desired outcomes over all potential outcomes, and you’ll be able to tackle these kinds of ACT math questions in no time. What’s Next? Now that you've stacked the odds in your favor on your probability questions, it's time to make sure you're caught up with the rest of your ACT math topics. We've got guides on all your individual math needs, from trigonometry to slopes and more. Wondering how your score stacks up? See what makes a "good" score and how you can get the most out of your studying time to reach your target goal. Running out of time on the ACT? Look to our guide on how to maximize your time and your score in the hour allotted. Want to get a perfect score? Check out how to get a perfect score on the ACT math, written by a 36-scorer. Want to improve your ACT score by 4 points? Check out our best-in-class online ACT prep program. 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